simplipy-complex-branch
Complex flavour: the rule over the complex numbers with principal branches; tests branch cuts of log, pow and the inverse functions.
Case set ba9dc4def822
sp-acj4-0-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \operatorname{arcosh}\left(\cosh x\right) \overset{?}{=} \left|x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[0]: acosh cosh _0 -> abs _0
Lean proof: included
(case sp-acj4-0-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[0]: acosh cosh _0 -> abs _0") (var x complex) (ask (= (acosh (cosh x)) (abs x))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-102-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \frac{\left|x\right|}{6} = \left|\frac{x}{-6}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6 -> abs / _0 (-6)
Lean proof: included
(case sp-acj4-102-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6 -> abs / _0 (-6)") (var x complex) (ask (= (/ (abs x) 6) (abs (/ x -6)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-108-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \left|x\right| \neq 0\)
Claim
\(\displaystyle \frac{\left|x\right|}{x} \overset{?}{=} \frac{x}{\left|x\right|}\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0 -> / _0 abs _0
Lean proof: included
(case sp-acj4-108-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0 -> / _0 abs _0") (var x complex) (assume (nonzero x)) (assume (nonzero (abs x))) (ask (= (/ (abs x) x) (/ x (abs x)))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")))
sp-acj4-120-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{\pi}{\left|x\right|} = \left|\frac{\pi}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0 -> abs / np.pi _0
Lean proof: included
(case sp-acj4-120-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0 -> abs / np.pi _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ pi (abs x)) (abs (/ pi x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-130-complex
Given
\(x \in \mathbb{C},\quad \cos x \neq 0,\quad \frac{1}{\cos x} \neq 0\)
Claim
\(\displaystyle \left|\log\left(\cos x\right)\right| \overset{?}{=} \log\left(\frac{1}{\cos x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2\)
SimpliPy acj-4 (MIT) rules_real.json[130]: abs log cos _0 -> log inv cos _0
Lean proof: included
(case sp-acj4-130-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[130]: abs log cos _0 -> log inv cos _0") (var x complex) (assume (nonzero (cos x))) (assume (nonzero (/ (cos x)))) (ask (= (abs (log (cos x))) (log (/ (cos x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x 2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-131-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \left|\log\left(\cosh x\right)\right| \overset{?}{=} \log\left(\cosh x\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0 -> log cosh _0
Lean proof: included
(case sp-acj4-131-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0 -> log cosh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (abs (log (cosh x))) (log (cosh x)))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-132-complex
Given
\(x \in \mathbb{C},\quad \sin x \neq 0\)
Claim
\(\displaystyle \left|\log\left(\sin x\right)\right| \overset{?}{=} -\log\left(\sin x\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0 -> neg log sin _0
Lean proof: included
(case sp-acj4-132-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0 -> neg log sin _0") (var x complex) (assume (nonzero (sin x))) (ask (= (abs (log (sin x))) (- (log (sin x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-133-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \tanh x \neq 0\)
Claim
\(\displaystyle \left|\log\left(\tanh x\right)\right| \overset{?}{=} -\log\left(\tanh x\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0 -> neg log tanh _0
Lean proof: included
(case sp-acj4-133-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0 -> neg log tanh _0") (var x complex) (assume (nonzero (cosh x))) (assume (nonzero (tanh x))) (ask (= (abs (log (tanh x))) (- (log (tanh x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-137-complex
Given
\(x \in \mathbb{C},\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0\)
Claim
\(\displaystyle \left|x^{\pi}\right| \overset{?}{=} x^{\pi}\)
Expected
contingent true at \(x = 0\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi -> pow _0 np.pi
Lean proof: included
(case sp-acj4-137-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi -> pow _0 np.pi") (var x complex) (assume (or (nonzero x) (> (re pi) 0))) (ask (= (abs (pow x pi)) (pow x pi))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-147-complex
Given
\(x \in \mathbb{C},\quad \cosh\left(\operatorname{arsinh}\left(x\right)\right) \neq 0,\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \left|\tanh\left(\operatorname{arsinh}\left(x\right)\right)\right| \overset{?}{=} \cos\left(\arctan\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = 1 + i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 1 + i\)
SimpliPy acj-4 (MIT) rules_real.json[147]: abs tanh asinh _0 -> cos atan inv _0
Lean proof: included
(case sp-acj4-147-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[147]: abs tanh asinh _0 -> cos atan inv _0") (var x complex) (assume (nonzero (cosh (asinh x)))) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (abs (tanh (asinh x))) (cos (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_log_inv_pi_I']")) (witness false (x (+ 1 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-151-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \arccos\left(\cos\left(\arcsin x\right)\right) \overset{?}{=} \left|\arcsin x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = 2\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2\)
SimpliPy acj-4 (MIT) rules_real.json[151]: acos cos asin _0 -> abs asin _0
Lean proof: included
(case sp-acj4-151-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[151]: acos cos asin _0 -> abs asin _0") (var x complex) (ask (= (acos (cos (asin x))) (abs (asin x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x 2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-153-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \arccos\left(\cos\left(\cos x\right)\right) \overset{?}{=} \left|\cos x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[153]: acos cos cos _0 -> abs cos _0
Lean proof: included
(case sp-acj4-153-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[153]: acos cos cos _0 -> abs cos _0") (var x complex) (ask (= (acos (cos (cos x))) (abs (cos x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-154-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \arccos\left(\cos\left(\sin x\right)\right) \overset{?}{=} \left|\sin x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[154]: acos cos sin _0 -> abs sin _0
Lean proof: included
(case sp-acj4-154-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[154]: acos cos sin _0 -> abs sin _0") (var x complex) (ask (= (acos (cos (sin x))) (abs (sin x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-165-complex
Given
\(x \in \mathbb{C},\quad x \neq i,\quad x \neq -i,\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \arcsin\left(\cos\left(\arctan x\right)\right) \overset{?}{=} \left|\arctan\left(\frac{1}{x}\right)\right|\)
Expected
contingent true at \(x = 1\); false at \(x = 1 + i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 1 + i\)
SimpliPy acj-4 (MIT) rules_real.json[165]: asin cos atan _0 -> abs atan inv _0
Lean proof: included
(case sp-acj4-165-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[165]: asin cos atan _0 -> abs atan inv _0") (var x complex) (assume (!= x i)) (assume (!= x (- i))) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (asin (cos (atan x))) (abs (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (+ 1 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-173-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \arcsin\left(\tanh\left(\left|x\right|\right)\right) \overset{?}{=} \arccos\left(\frac{1}{\cosh x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[173]: asin tanh abs _0 -> acos inv cosh _0
Lean proof: included
(case sp-acj4-173-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[173]: asin tanh abs _0 -> acos inv cosh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (asin (tanh (abs x))) (acos (/ (cosh x))))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-176-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \arctan\left(\left|\sinh x\right|\right) \overset{?}{=} \arccos\left(\frac{1}{\cosh x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[176]: atan abs sinh _0 -> acos inv cosh _0
Lean proof: included
(case sp-acj4-176-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[176]: atan abs sinh _0 -> acos inv cosh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (atan (abs (sinh x))) (acos (/ (cosh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-177-complex
Given
\(x \in \mathbb{C},\quad \cos x \neq 0\)
Claim
\(\displaystyle \arctan\left(\left|\tan x\right|\right) \overset{?}{=} \left|\arcsin\left(\sin x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[177]: atan abs tan _0 -> abs asin sin _0
Lean proof: included
(case sp-acj4-177-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[177]: atan abs tan _0 -> abs asin sin _0") (var x complex) (assume (nonzero (cos x))) (ask (= (atan (abs (tan x))) (abs (asin (sin x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-180-complex
Given
\(x \in \mathbb{C},\quad \sinh\left(\operatorname{arcosh}\left(x\right)\right) \neq i,\quad \sinh\left(\operatorname{arcosh}\left(x\right)\right) \neq -i,\quad \cosh\left(\operatorname{arcosh}\left(x\right)\right) \neq 0\)
Claim
\(\displaystyle \arctan\left(\sinh\left(\operatorname{arcosh}\left(x\right)\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\operatorname{arcosh}\left(x\right)\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[180]: atan sinh acosh _0 -> asin tanh acosh _0
Lean proof: included
(case sp-acj4-180-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[180]: atan sinh acosh _0 -> asin tanh acosh _0") (var x complex) (assume (!= (sinh (acosh x)) i)) (assume (!= (sinh (acosh x)) (- i))) (assume (nonzero (cosh (acosh x)))) (ask (= (atan (sinh (acosh x))) (asin (tanh (acosh x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234]; exact AE.w1_c180")))
sp-acj4-183-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \sinh\left(\log x\right) \neq i,\quad \sinh\left(\log x\right) \neq -i,\quad \cosh\left(\log x\right) \neq 0\)
Claim
\(\displaystyle \arctan\left(\sinh\left(\log x\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\log x\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[183]: atan sinh log _0 -> asin tanh log _0
Lean proof: included
(case sp-acj4-183-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[183]: atan sinh log _0 -> asin tanh log _0") (var x complex) (assume (nonzero x)) (assume (!= (sinh (log x)) i)) (assume (!= (sinh (log x)) (- i))) (assume (nonzero (cosh (log x)))) (ask (= (atan (sinh (log x))) (asin (tanh (log x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234]; exact AE.w1_c183")))
sp-acj4-189-complex
Given
\(x \in \mathbb{C},\quad \cos\left(\arcsin x\right) \neq 1,\quad \cos\left(\arcsin x\right) \neq -1,\quad x \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\cos\left(\arcsin x\right)\right) \overset{?}{=} \operatorname{arcosh}\left(\left|\frac{1}{x}\right|\right)\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[189]: atanh cos asin _0 -> acosh abs inv _0
Lean proof: included
(case sp-acj4-189-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[189]: atanh cos asin _0 -> acosh abs inv _0") (var x complex) (assume (!= (cos (asin x)) 1)) (assume (!= (cos (asin x)) -1)) (assume (nonzero x)) (ask (= (atanh (cos (asin x))) (acosh (abs (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-190-complex
Given
\(x \in \mathbb{C},\quad x \neq i,\quad x \neq -i,\quad \cos\left(\arctan x\right) \neq 1,\quad \cos\left(\arctan x\right) \neq -1,\quad x \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\cos\left(\arctan x\right)\right) \overset{?}{=} \left|\operatorname{arsinh}\left(\frac{1}{x}\right)\right|\)
Expected
contingent true at \(x = 1\); false at \(x = \frac{1}{2} \, i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = \frac{1}{2} \, i\)
SimpliPy acj-4 (MIT) rules_real.json[190]: atanh cos atan _0 -> abs asinh inv _0
Lean proof: included
(case sp-acj4-190-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[190]: atanh cos atan _0 -> abs asinh inv _0") (var x complex) (assume (!= x i)) (assume (!= x (- i))) (assume (!= (cos (atan x)) 1)) (assume (!= (cos (atan x)) -1)) (assume (nonzero x)) (ask (= (atanh (cos (atan x))) (abs (asinh (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 1/2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-191-complex
Given
\(x \in \mathbb{C},\quad \sin\left(\arccos x\right) \neq 1,\quad \sin\left(\arccos x\right) \neq -1,\quad x \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\sin\left(\arccos x\right)\right) \overset{?}{=} \operatorname{arcosh}\left(\left|\frac{1}{x}\right|\right)\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[191]: atanh sin acos _0 -> acosh abs inv _0
Lean proof: included
(case sp-acj4-191-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[191]: atanh sin acos _0 -> acosh abs inv _0") (var x complex) (assume (!= (sin (acos x)) 1)) (assume (!= (sin (acos x)) -1)) (assume (nonzero x)) (ask (= (atanh (sin (acos x))) (acosh (abs (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-20-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \sinh\left(\left|x\right|\right) \overset{?}{=} \left|\sinh x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[20] + rules.json[15]: sinh abs _0 -> abs sinh _0
Lean proof: included
(case sp-acj4-20-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[20] + rules.json[15]: sinh abs _0 -> abs sinh _0") (var x complex) (ask (= (sinh (abs x)) (abs (sinh x)))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-203-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0\)
Claim
\(\displaystyle \cos\left(\log\left(\frac{1}{x}\right)\right) \overset{?}{=} \cos\left(\log x\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[203]: cos log inv _0 -> cos log _0
Lean proof: included
(case sp-acj4-203-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[203]: cos log inv _0 -> cos log _0") (var x complex) (assume (nonzero x)) (assume (nonzero (/ x))) (ask (= (cos (log (/ x))) (cos (log x)))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-213-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{-10}{x} \neq 0\)
Claim
\(\displaystyle \frac{1}{\frac{-10}{x}} = \frac{x}{-10}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[213] + rules.json[145]: inv / (-10) !0 -> / !0 (-10)
Lean proof: included
(case sp-acj4-213-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[213] + rules.json[145]: inv / (-10) !0 -> / !0 (-10)") (var x complex) (assume (nonzero x)) (assume (nonzero (/ -10 x))) (ask (= (/ (/ -10 x)) (/ x -10))) (expect true) (lean-proof "simp_all"))
sp-acj4-216-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{-4}{x} \neq 0\)
Claim
\(\displaystyle \frac{1}{\frac{-4}{x}} = \frac{x}{-4}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[216] + rules.json[148]: inv / (-4) !0 -> / !0 (-4)
Lean proof: included
(case sp-acj4-216-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[216] + rules.json[148]: inv / (-4) !0 -> / !0 (-4)") (var x complex) (assume (nonzero x)) (assume (nonzero (/ -4 x))) (ask (= (/ (/ -4 x)) (/ x -4))) (expect true) (lean-proof "simp_all"))
sp-acj4-22-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \tanh\left(\left|x\right|\right) \overset{?}{=} \left|\tanh x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0 -> abs tanh _0
Lean proof: included
(case sp-acj4-22-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0 -> abs tanh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (tanh (abs x)) (abs (tanh x)))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-234-complex
Given
\(x \in \mathbb{C},\quad x \neq 0 \lor \operatorname{Re}\left(x\right) > 0,\quad x^{x} \neq 0,\quad x \neq 0,\quad \frac{1}{x} \neq 0 \lor \operatorname{Re}\left(x\right) > 0\)
Claim
\(\displaystyle \frac{1}{x^{x}} \overset{?}{=} \frac{1}{x}^{x}\)
Expected
contingent true at \(x = 1\); false at \(x = -\frac{1}{2}\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -\frac{1}{2}\)
SimpliPy acj-4 (MIT) rules_real.json[234]: inv pow _0 _0 -> pow inv _0 _0
Lean proof: included
(case sp-acj4-234-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[234]: inv pow _0 _0 -> pow inv _0 _0") (var x complex) (assume (or (nonzero x) (> (re x) 0))) (assume (nonzero (pow x x))) (assume (nonzero x)) (assume (or (nonzero (/ x)) (> (re x) 0))) (ask (= (/ (pow x x)) (pow (/ x) x))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1/2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-252-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle -\left|\sinh x\right| \overset{?}{=} \sinh\left(-\left|x\right|\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0 -> sinh neg abs _0
Lean proof: included
(case sp-acj4-252-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0 -> sinh neg abs _0") (var x complex) (ask (= (- (abs (sinh x))) (sinh (- (abs x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-253-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle -\left|\tanh x\right| \overset{?}{=} \tanh\left(-\left|x\right|\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0 -> tanh neg abs _0
Lean proof: included
(case sp-acj4-253-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0 -> tanh neg abs _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (- (abs (tanh x))) (tanh (- (abs x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-260-complex
Given
\(x \in \mathbb{C},\quad \cos x \neq 0,\quad \frac{1}{\cos x} \neq 0\)
Claim
\(\displaystyle -\log\left(\cos x\right) \overset{?}{=} \log\left(\frac{1}{\cos x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2\)
SimpliPy acj-4 (MIT) rules_real.json[260]: neg log cos _0 -> log inv cos _0
Lean proof: included
(case sp-acj4-260-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[260]: neg log cos _0 -> log inv cos _0") (var x complex) (assume (nonzero (cos x))) (assume (nonzero (/ (cos x)))) (ask (= (- (log (cos x))) (log (/ (cos x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x 2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-261-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \frac{1}{\cosh x} \neq 0\)
Claim
\(\displaystyle -\log\left(\cosh x\right) \overset{?}{=} \log\left(\frac{1}{\cosh x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[261]: neg log cosh _0 -> log inv cosh _0
Lean proof: included
(case sp-acj4-261-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[261]: neg log cosh _0 -> log inv cosh _0") (var x complex) (assume (nonzero (cosh x))) (assume (nonzero (/ (cosh x)))) (ask (= (- (log (cosh x))) (log (/ (cosh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-266-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0\)
Claim
\(\displaystyle -\sin\left(\log x\right) \overset{?}{=} \sin\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[266]: neg sin log _0 -> sin log inv _0
Lean proof: included
(case sp-acj4-266-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[266]: neg sin log _0 -> sin log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (/ x))) (ask (= (- (sin (log x))) (sin (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-267-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \cos\left(\log x\right) \neq 0,\quad \frac{1}{x} \neq 0,\quad \cos\left(\log\left(\frac{1}{x}\right)\right) \neq 0\)
Claim
\(\displaystyle -\tan\left(\log x\right) \overset{?}{=} \tan\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[267]: neg tan log _0 -> tan log inv _0
Lean proof: included
(case sp-acj4-267-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[267]: neg tan log _0 -> tan log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (cos (log x)))) (assume (nonzero (/ x))) (assume (nonzero (cos (log (/ x))))) (ask (= (- (tan (log x))) (tan (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-274-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \left|x\right|^{-2} \overset{?}{=} x^{-2}\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[274] + rules.json[183]: pow abs _0 (-2) -> pow _0 (-2)
Lean proof: included
(case sp-acj4-274-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[274] + rules.json[183]: pow abs _0 (-2) -> pow _0 (-2)") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (pow (abs x) -2) (pow x -2))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-278-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \left|x\right|^{-6} \overset{?}{=} x^{-6}\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[278] + rules.json[187]: pow abs _0 (-6) -> pow _0 (-6)
Lean proof: included
(case sp-acj4-278-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[278] + rules.json[187]: pow abs _0 (-6) -> pow _0 (-6)") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (pow (abs x) -6) (pow x -6))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-283-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right|^{2} \overset{?}{=} x \, x\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2 -> * _0 _0
Lean proof: included
(case sp-acj4-283-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2 -> * _0 _0") (var x complex) (ask (= (pow (abs x) 2) (* x x))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")))
sp-acj4-284-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right|^{3} = \left|x^{3}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[284] + rules.json[193]: pow abs _0 3 -> abs pow _0 3
Lean proof: included
(case sp-acj4-284-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[284] + rules.json[193]: pow abs _0 3 -> abs pow _0 3") (var x complex) (ask (= (pow (abs x) 3) (abs (pow x 3)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-288-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right|^{7} = \left|x^{7}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[288] + rules.json[197]: pow abs _0 7 -> abs pow _0 7
Lean proof: included
(case sp-acj4-288-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[288] + rules.json[197]: pow abs _0 7 -> abs pow _0 7") (var x complex) (ask (= (pow (abs x) 7) (abs (pow x 7)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-30-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle 4 \, \left|x\right| = \left|-4 \, x\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[30] + rules.json[23]: * 4 abs _0 -> abs * (-4) _0
Lean proof: included
(case sp-acj4-30-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[30] + rules.json[23]: * 4 abs _0 -> abs * (-4) _0") (var x complex) (ask (= (* 4 (abs x)) (abs (* -4 x)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-334-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0 \lor \operatorname{Re}\left(e^{1}\right) > 0\)
Claim
\(\displaystyle \left(e^{x}\right)^{e^{1}} \overset{?}{=} e^{e^{1} \, x}\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[334]: pow exp _0 np.e -> exp * np.e _0
Lean proof: included
(case sp-acj4-334-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[334]: pow exp _0 np.e -> exp * np.e _0") (var x complex) (assume (or (nonzero (exp x)) (> (re (exp 1)) 0))) (ask (= (pow (exp x) (exp 1)) (exp (* (exp 1) x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-335-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0\)
Claim
\(\displaystyle \left(e^{x}\right)^{\pi} \overset{?}{=} e^{\pi \, x}\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[335]: pow exp _0 np.pi -> exp * np.pi _0
Lean proof: included
(case sp-acj4-335-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[335]: pow exp _0 np.pi -> exp * np.pi _0") (var x complex) (assume (or (nonzero (exp x)) (> (re pi) 0))) (ask (= (pow (exp x) pi) (exp (* pi x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-336-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0 \lor \operatorname{Re}\left(x\right) > 0\)
Claim
\(\displaystyle \left(e^{x}\right)^{x} \overset{?}{=} e^{x \, x}\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[336]: pow exp _0 _0 -> exp * _0 _0
Lean proof: included
(case sp-acj4-336-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[336]: pow exp _0 _0 -> exp * _0 _0") (var x complex) (assume (or (nonzero (exp x)) (> (re x) 0))) (ask (= (pow (exp x) x) (exp (* x x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-343-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \frac{1}{\pi}^{x} = \pi^{-x}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[343]: pow inv np.pi _0 -> pow np.pi neg _0
Lean proof: included
(case sp-acj4-343-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[343]: pow inv np.pi _0 -> pow np.pi neg _0") (var x complex) (ask (= (pow (/ pi) x) (pow pi (- x)))) (expect true) (lean-proof "refine ⟨by simp [Real.pi_ne_zero], ?_⟩; simp only [AE.pi_def, AE.inv_def]; exact AE.pow_inv_base Real.pi_pos x"))
sp-acj4-344-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0 \lor \operatorname{Re}\left(x\right) > 0,\quad x \neq 0 \lor \operatorname{Re}\left(-x\right) > 0\)
Claim
\(\displaystyle \frac{1}{x}^{x} \overset{?}{=} x^{-x}\)
Expected
contingent true at \(x = 1\); false at \(x = -\frac{1}{2}\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -\frac{1}{2}\)
SimpliPy acj-4 (MIT) rules_real.json[344]: pow inv _0 _0 -> pow _0 neg _0
Lean proof: included
(case sp-acj4-344-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[344]: pow inv _0 _0 -> pow _0 neg _0") (var x complex) (assume (nonzero x)) (assume (or (nonzero (/ x)) (> (re x) 0))) (assume (or (nonzero x) (> (re (- x)) 0))) (ask (= (pow (/ x) x) (pow x (- x)))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1/2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-368-complex
Given
\(x \in \mathbb{C},\quad \arccos x \in \mathbb{R},\quad \arccos x \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(\arccos x, -5\right) = \left(\arccos x\right)^{-\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5) -> pow acos _0 -0.2
Lean proof: included
(case sp-acj4-368-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5) -> pow acos _0 -0.2") (var x complex) (assume (real (acos x))) (assume (nonzero (acos x))) (ask (= (surd (acos x) -5) (pow (acos x) -1/5))) (expect true) (lean-proof "have hi : (AE.acos x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acos_re_nonneg hi; refine ⟨⟨⟨AE.acos.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨-5, by norm_num⟩ : ∃ n : ℤ, (-5:ℂ) = n), by norm_num, Or.inl hr, Or.inl h1.2⟩⟩, AE.acos.D_true x, (AE.pow.D_iff _ _).2 (Or.inl h1.2)⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]; norm_num"))
sp-acj4-369-complex
Given
\(x \in \mathbb{C},\quad \arccos x \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(\arccos x, 5\right) = \left(\arccos x\right)^{\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5 -> pow acos _0 0.2
Lean proof: included
(case sp-acj4-369-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5 -> pow acos _0 0.2") (var x complex) (assume (real (acos x))) (ask (= (surd (acos x) 5) (pow (acos x) 1/5))) (expect true) (lean-proof "have hi : (AE.acos x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acos_re_nonneg hi; refine ⟨⟨⟨AE.acos.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨5, by norm_num⟩ : ∃ n : ℤ, (5:ℂ) = n), by norm_num, Or.inl hr, Or.inr (by norm_num)⟩⟩, AE.acos.D_true x, (AE.pow.D_iff _ _).2 (Or.inr (by norm_num))⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]"))
sp-acj4-370-complex
Given
\(x \in \mathbb{C},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), -5\right) = \operatorname{arcosh}\left(x\right)^{-\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _0 -0.2
Lean proof: included
(case sp-acj4-370-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _0 -0.2") (var x complex) (assume (real (acosh x))) (assume (nonzero (acosh x))) (ask (= (surd (acosh x) -5) (pow (acosh x) -1/5))) (expect true) (lean-proof "have hi : (AE.acosh x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acosh_re_nonneg hi; refine ⟨⟨⟨AE.acosh.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨-5, by norm_num⟩ : ∃ n : ℤ, (-5:ℂ) = n), by norm_num, Or.inl hr, Or.inl h1.2⟩⟩, AE.acosh.D_true x, (AE.pow.D_iff _ _).2 (Or.inl h1.2)⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]; norm_num"))
sp-acj4-371-complex
Given
\(x \in \mathbb{C},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), 5\right) = \operatorname{arcosh}\left(x\right)^{\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5 -> pow acosh _0 0.2
Lean proof: included
(case sp-acj4-371-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5 -> pow acosh _0 0.2") (var x complex) (assume (real (acosh x))) (ask (= (surd (acosh x) 5) (pow (acosh x) 1/5))) (expect true) (lean-proof "have hi : (AE.acosh x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acosh_re_nonneg hi; refine ⟨⟨⟨AE.acosh.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨5, by norm_num⟩ : ∃ n : ℤ, (5:ℂ) = n), by norm_num, Or.inl hr, Or.inr (by norm_num)⟩⟩, AE.acosh.D_true x, (AE.pow.D_iff _ _).2 (Or.inr (by norm_num))⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]"))
sp-acj4-372-complex
Given
\(x \in \mathbb{C},\quad \cosh x \in \mathbb{R},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(\cosh x, -5\right) \overset{?}{=} \left(\cosh x\right)^{-\frac{1}{5}}\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5) -> pow cosh _0 -0.2
Lean proof: included
(case sp-acj4-372-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5) -> pow cosh _0 -0.2") (var x complex) (assume (real (cosh x))) (assume (nonzero (cosh x))) (ask (= (surd (cosh x) -5) (pow (cosh x) -1/5))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-373-complex
Given
\(x \in \mathbb{C},\quad \cosh x \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(\cosh x, 5\right) \overset{?}{=} \left(\cosh x\right)^{\frac{1}{5}}\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5 -> pow cosh _0 0.2
Lean proof: included
(case sp-acj4-373-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5 -> pow cosh _0 0.2") (var x complex) (assume (real (cosh x))) (ask (= (surd (cosh x) 5) (pow (cosh x) 1/5))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-387-complex
Given
\(x \in \mathbb{C},\quad e^{x} \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(e^{x}, 5\right) \overset{?}{=} e^{\frac{x}{5}}\)
Expected
contingent true at \(x = 0\); false at \(x = \pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = \pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[387]: rootn exp _0 5 -> exp / _0 5
Lean proof: included
(case sp-acj4-387-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[387]: rootn exp _0 5 -> exp / _0 5") (var x complex) (assume (real (exp x))) (ask (= (surd (exp x) 5) (exp (/ x 5)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-389-complex
Given
\(x \in \mathbb{C},\quad e^{x} \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(e^{x}, 7\right) \overset{?}{=} e^{\frac{x}{7}}\)
Expected
contingent true at \(x = 0\); false at \(x = \pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = \pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[389]: rootn exp _0 7 -> exp / _0 7
Lean proof: included
(case sp-acj4-389-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[389]: rootn exp _0 7 -> exp / _0 7") (var x complex) (assume (real (exp x))) (ask (= (surd (exp x) 7) (exp (/ x 7)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-396-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \sin\left(\left|\sin x\right|\right) \overset{?}{=} \left|\sin\left(\sin x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0 -> abs sin sin _0
Lean proof: included
(case sp-acj4-396-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0 -> abs sin sin _0") (var x complex) (ask (= (sin (abs (sin x))) (abs (sin (sin x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-40-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right| \, \pi = \left|\pi \, x\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[40] + rules.json[33]: * abs _0 np.pi -> abs * np.pi _0
Lean proof: included
(case sp-acj4-40-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[40] + rules.json[33]: * abs _0 np.pi -> abs * np.pi _0") (var x complex) (ask (= (* (abs x) pi) (abs (* pi x)))) (expect true) (lean-proof "simp [AE.abs_eq]; ring"))
sp-acj4-403-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0\)
Claim
\(\displaystyle \sin\left(-\log x\right) \overset{?}{=} \sin\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[403]: sin neg log _0 -> sin log inv _0
Lean proof: included
(case sp-acj4-403-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[403]: sin neg log _0 -> sin log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (/ x))) (ask (= (sin (- (log x))) (sin (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-404-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \sin\left(\tanh\left(\left|x\right|\right)\right) \overset{?}{=} \left|\sin\left(\tanh x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0 -> abs sin tanh _0
Lean proof: included
(case sp-acj4-404-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0 -> abs sin tanh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (sin (tanh (abs x))) (abs (sin (tanh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-414-complex
Given
\(x \in \mathbb{C},\quad \cos\left(\left|\sin x\right|\right) \neq 0,\quad \cos\left(\sin x\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\left|\sin x\right|\right) \overset{?}{=} \left|\tan\left(\sin x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0 -> abs tan sin _0
Lean proof: included
(case sp-acj4-414-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0 -> abs tan sin _0") (var x complex) (assume (nonzero (cos (abs (sin x))))) (assume (nonzero (cos (sin x)))) (ask (= (tan (abs (sin x))) (abs (tan (sin x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-415-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \cos\left(\left|\tanh x\right|\right) \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\left|\tanh x\right|\right) \overset{?}{=} \left|\tan\left(\tanh x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0 -> abs tan tanh _0
Lean proof: included
(case sp-acj4-415-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0 -> abs tan tanh _0") (var x complex) (assume (nonzero (cosh x))) (assume (nonzero (cos (abs (tanh x))))) (assume (nonzero (cos (tanh x)))) (ask (= (tan (abs (tanh x))) (abs (tan (tanh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-420-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \cos\left(-\log x\right) \neq 0,\quad \frac{1}{x} \neq 0,\quad \cos\left(\log\left(\frac{1}{x}\right)\right) \neq 0\)
Claim
\(\displaystyle \tan\left(-\log x\right) \overset{?}{=} \tan\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[420]: tan neg log _0 -> tan log inv _0
Lean proof: included
(case sp-acj4-420-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[420]: tan neg log _0 -> tan log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (cos (- (log x))))) (assume (nonzero (/ x))) (assume (nonzero (cos (log (/ x))))) (ask (= (tan (- (log x))) (tan (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-421-complex
Given
\(x \in \mathbb{C},\quad \cos\left(\tanh\left(\left|x\right|\right)\right) \neq 0,\quad \cosh x \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\tanh\left(\left|x\right|\right)\right) \overset{?}{=} \left|\tan\left(\tanh x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0 -> abs tan tanh _0
Lean proof: included
(case sp-acj4-421-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0 -> abs tan tanh _0") (var x complex) (assume (nonzero (cos (tanh (abs x))))) (assume (nonzero (cosh x))) (assume (nonzero (cos (tanh x)))) (ask (= (tan (tanh (abs x))) (abs (tan (tanh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-422-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \tanh\left(\left|\operatorname{arsinh}\left(x\right)\right|\right) \overset{?}{=} \cos\left(\arctan\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[422]: tanh abs asinh _0 -> cos atan inv _0
Lean proof: included
(case sp-acj4-422-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[422]: tanh abs asinh _0 -> cos atan inv _0") (var x complex) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (tanh (abs (asinh x))) (cos (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-423-complex
Given
\(x \in \mathbb{C},\quad \cosh\left(\operatorname{arcosh}\left(\left|x\right|\right)\right) \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \tanh\left(\operatorname{arcosh}\left(\left|x\right|\right)\right) \overset{?}{=} \cos\left(\arcsin\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[423]: tanh acosh abs _0 -> cos asin inv _0
Lean proof: included
(case sp-acj4-423-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[423]: tanh acosh abs _0 -> cos asin inv _0") (var x complex) (assume (nonzero (cosh (acosh (abs x))))) (assume (nonzero x)) (ask (= (tanh (acosh (abs x))) (cos (asin (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-424-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \tanh\left(\operatorname{arsinh}\left(\left|x\right|\right)\right) \overset{?}{=} \cos\left(\arctan\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[424]: tanh asinh abs _0 -> cos atan inv _0
Lean proof: included
(case sp-acj4-424-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[424]: tanh asinh abs _0 -> cos atan inv _0") (var x complex) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (tanh (asinh (abs x))) (cos (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-430-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(x\right) > 0,\quad x^{x} \neq 0\)
Claim
\(\displaystyle \log x \, x \overset{?}{=} \log\left(x^{x}\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[430]: * log ?0 ?0 -> log pow ?0 ?0
Lean proof: included
(case sp-acj4-430-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[430]: * log ?0 ?0 -> log pow ?0 ?0") (var x complex) (assume (nonzero x)) (assume (or (nonzero x) (> (re x) 0))) (assume (nonzero (pow x x))) (ask (= (* (log x) x) (log (pow x x)))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-433-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad -\log x \neq i,\quad -\log x \neq -i,\quad \frac{1}{x} \neq 0,\quad \log\left(\frac{1}{x}\right) \neq i,\quad \log\left(\frac{1}{x}\right) \neq -i\)
Claim
\(\displaystyle \arctan\left(-\log x\right) \overset{?}{=} \arctan\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[433]: atan neg log !0 -> atan log inv !0
Lean proof: included
(case sp-acj4-433-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[433]: atan neg log !0 -> atan log inv !0") (var x complex) (assume (nonzero x)) (assume (!= (- (log x)) i)) (assume (!= (- (log x)) (- i))) (assume (nonzero (/ x))) (assume (!= (log (/ x)) i)) (assume (!= (log (/ x)) (- i))) (ask (= (atan (- (log x))) (atan (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234]; exact AE.w1_c433")))
sp-acj4-442-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \log x \neq 0,\quad \frac{1}{\log x} \neq 0\)
Claim
\(\displaystyle -\log\left(\log x\right) \overset{?}{=} \log\left(\frac{1}{\log x}\right)\)
Expected
contingent true at \(x = -1\); false at \(x = \frac{1}{2}\)
equal at some admissible points, different at others
witness (true): \(x = -1\)
witness (false): \(x = \frac{1}{2}\)
SimpliPy acj-4 (MIT) rules_real.json[442]: neg log log ?0 -> log inv log ?0
Lean proof: included
(case sp-acj4-442-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[442]: neg log log ?0 -> log inv log ?0") (var x complex) (assume (nonzero x)) (assume (nonzero (log x))) (assume (nonzero (/ (log x)))) (ask (= (- (log (log x))) (log (/ (log x))))) (expect contingent) (witness true (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x 1/2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-446-complex
Given
\(x \in \mathbb{C},\quad x \neq i,\quad x \neq -i,\quad \cos\left(\left|\arctan x\right|\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\left|\arctan x\right|\right) \overset{?}{=} \left|x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = \frac{1}{2} \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = \frac{1}{2} \, i\)
SimpliPy acj-4 (MIT) rules_real.json[446]: tan abs atan !0 -> abs !0
Lean proof: included
(case sp-acj4-446-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[446]: tan abs atan !0 -> abs !0") (var x complex) (assume (!= x i)) (assume (!= x (- i))) (assume (nonzero (cos (abs (atan x))))) (ask (= (tan (abs (atan x))) (abs x))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x (* 1/2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-450-complex
Given
\(x \in \mathbb{C},\quad x \neq 0\)
Claim
\(\displaystyle \frac{x}{x} = 1\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[450]: / $0 $0 -> 1
Lean proof: included
(case sp-acj4-450-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[450]: / $0 $0 -> 1") (var x complex) (assume (nonzero x)) (ask (= (/ x x) 1)) (expect true) (lean-proof "simp_all"))
sp-acj4-451-complex
Given
\(x \in \mathbb{C},\quad x \neq 0\)
Claim
\(\displaystyle \frac{0}{x} = 0\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[451]: / 0 $0 -> 0
Lean proof: included
(case sp-acj4-451-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[451]: / 0 $0 -> 0") (var x complex) (assume (nonzero x)) (ask (= (/ 0 x) 0)) (expect true) (lean-proof "simp_all"))
sp-acj4-50-complex
Given
\(x \in \mathbb{C},\quad 6^{x} \neq 0\)
Claim
\(\displaystyle \log 6 \, x \overset{?}{=} \log\left(6^{x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[50]: * log 6 _0 -> log pow 6 _0
Lean proof: included
(case sp-acj4-50-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[50]: * log 6 _0 -> log pow 6 _0") (var x complex) (assume (nonzero (pow 6 x))) (ask (= (* (log 6) x) (log (pow 6 x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-51-complex
Given
\(x \in \mathbb{C},\quad 7^{x} \neq 0\)
Claim
\(\displaystyle \log 7 \, x \overset{?}{=} \log\left(7^{x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[51]: * log 7 _0 -> log pow 7 _0
Lean proof: included
(case sp-acj4-51-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[51]: * log 7 _0 -> log pow 7 _0") (var x complex) (assume (nonzero (pow 7 x))) (ask (= (* (log 7) x) (log (pow 7 x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-55-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(e^{1}\right) > 0,\quad x^{e^{1}} \neq 0\)
Claim
\(\displaystyle \log x \, e^{1} \overset{?}{=} \log\left(x^{e^{1}}\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[55]: * log _0 np.e -> log pow _0 np.e
Lean proof: included
(case sp-acj4-55-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[55]: * log _0 np.e -> log pow _0 np.e") (var x complex) (assume (nonzero x)) (assume (or (nonzero x) (> (re (exp 1)) 0))) (assume (nonzero (pow x (exp 1)))) (ask (= (* (log x) (exp 1)) (log (pow x (exp 1))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-56-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0,\quad x^{\pi} \neq 0\)
Claim
\(\displaystyle \log x \, \pi \overset{?}{=} \log\left(x^{\pi}\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[56]: * log _0 np.pi -> log pow _0 np.pi
Lean proof: included
(case sp-acj4-56-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[56]: * log _0 np.pi -> log pow _0 np.pi") (var x complex) (assume (nonzero x)) (assume (or (nonzero x) (> (re pi) 0))) (assume (nonzero (pow x pi))) (ask (= (* (log x) pi) (log (pow x pi)))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-6-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \tanh x \neq 1,\quad \tanh x \neq -1\)
Claim
\(\displaystyle \operatorname{artanh}\left(\tanh x\right) \overset{?}{=} x\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[6]: atanh tanh !0 -> !0
Lean proof: included
(case sp-acj4-6-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[6]: atanh tanh !0 -> !0") (var x complex) (assume (nonzero (cosh x))) (assume (!= (tanh x) 1)) (assume (!= (tanh x) -1)) (ask (= (atanh (tanh x)) x)) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-7-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{1}{\left|x\right|} = \left|\frac{1}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[7] + rules.json[4]: inv abs _0 -> abs inv _0
Lean proof: included
(case sp-acj4-7-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[7] + rules.json[4]: inv abs _0 -> abs inv _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ (abs x)) (abs (/ x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-9-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0\)
Claim
\(\displaystyle \log\left(e^{x}\right) \overset{?}{=} x\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[9]: log exp _0 -> _0
Lean proof: included
(case sp-acj4-9-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[9]: log exp _0 -> _0") (var x complex) (assume (nonzero (exp x))) (ask (= (log (exp x)) x)) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-90-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{3}{\left|x\right|} = \left|\frac{-3}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0 -> abs / (-3) _0
Lean proof: included
(case sp-acj4-90-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0 -> abs / (-3) _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ 3 (abs x)) (abs (/ -3 x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-94-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{7}{\left|x\right|} = \left|\frac{-7}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[94] + rules.json[73]: / 7 abs _0 -> abs / (-7) _0
Lean proof: included
(case sp-acj4-94-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[94] + rules.json[73]: / 7 abs _0 -> abs / (-7) _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ 7 (abs x)) (abs (/ -7 x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-99-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \frac{\left|x\right|}{3} = \left|\frac{x}{-3}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3 -> abs / _0 (-3)
Lean proof: included
(case sp-acj4-99-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3 -> abs / _0 (-3)") (var x complex) (ask (= (/ (abs x) 3) (abs (/ x -3)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
Case set 3ba0dfac4bca
sp-acj4-0-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \operatorname{arcosh}\left(\cosh x\right) \overset{?}{=} \left|x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[0]: acosh cosh _0 -> abs _0
Lean proof: included
(case sp-acj4-0-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[0]: acosh cosh _0 -> abs _0") (var x complex) (ask (= (acosh (cosh x)) (abs x))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-102-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \frac{\left|x\right|}{6} = \left|\frac{x}{-6}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6 -> abs / _0 (-6)
Lean proof: included
(case sp-acj4-102-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6 -> abs / _0 (-6)") (var x complex) (ask (= (/ (abs x) 6) (abs (/ x -6)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-108-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \left|x\right| \neq 0\)
Claim
\(\displaystyle \frac{\left|x\right|}{x} \overset{?}{=} \frac{x}{\left|x\right|}\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0 -> / _0 abs _0
Lean proof: included
(case sp-acj4-108-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0 -> / _0 abs _0") (var x complex) (assume (nonzero x)) (assume (nonzero (abs x))) (ask (= (/ (abs x) x) (/ x (abs x)))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")))
sp-acj4-120-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{\pi}{\left|x\right|} = \left|\frac{\pi}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0 -> abs / np.pi _0
Lean proof: included
(case sp-acj4-120-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0 -> abs / np.pi _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ pi (abs x)) (abs (/ pi x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-130-complex
Given
\(x \in \mathbb{C},\quad \cos x \neq 0,\quad \frac{1}{\cos x} \neq 0\)
Claim
\(\displaystyle \left|\log\left(\cos x\right)\right| \overset{?}{=} \log\left(\frac{1}{\cos x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2\)
SimpliPy acj-4 (MIT) rules_real.json[130]: abs log cos _0 -> log inv cos _0
Lean proof: included
(case sp-acj4-130-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[130]: abs log cos _0 -> log inv cos _0") (var x complex) (assume (nonzero (cos x))) (assume (nonzero (/ (cos x)))) (ask (= (abs (log (cos x))) (log (/ (cos x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x 2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-131-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \left|\log\left(\cosh x\right)\right| \overset{?}{=} \log\left(\cosh x\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0 -> log cosh _0
Lean proof: included
(case sp-acj4-131-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0 -> log cosh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (abs (log (cosh x))) (log (cosh x)))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-132-complex
Given
\(x \in \mathbb{C},\quad \sin x \neq 0\)
Claim
\(\displaystyle \left|\log\left(\sin x\right)\right| \overset{?}{=} -\log\left(\sin x\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0 -> neg log sin _0
Lean proof: included
(case sp-acj4-132-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0 -> neg log sin _0") (var x complex) (assume (nonzero (sin x))) (ask (= (abs (log (sin x))) (- (log (sin x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-133-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \tanh x \neq 0\)
Claim
\(\displaystyle \left|\log\left(\tanh x\right)\right| \overset{?}{=} -\log\left(\tanh x\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0 -> neg log tanh _0
Lean proof: included
(case sp-acj4-133-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0 -> neg log tanh _0") (var x complex) (assume (nonzero (cosh x))) (assume (nonzero (tanh x))) (ask (= (abs (log (tanh x))) (- (log (tanh x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-137-complex
Given
\(x \in \mathbb{C},\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0\)
Claim
\(\displaystyle \left|x^{\pi}\right| \overset{?}{=} x^{\pi}\)
Expected
contingent true at \(x = 0\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi -> pow _0 np.pi
Lean proof: included
(case sp-acj4-137-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi -> pow _0 np.pi") (var x complex) (assume (or (nonzero x) (> (re pi) 0))) (ask (= (abs (pow x pi)) (pow x pi))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-147-complex
Given
\(x \in \mathbb{C},\quad \cosh\left(\operatorname{arsinh}\left(x\right)\right) \neq 0,\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \left|\tanh\left(\operatorname{arsinh}\left(x\right)\right)\right| \overset{?}{=} \cos\left(\arctan\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = 1 + i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 1 + i\)
SimpliPy acj-4 (MIT) rules_real.json[147]: abs tanh asinh _0 -> cos atan inv _0
Lean proof: included
(case sp-acj4-147-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[147]: abs tanh asinh _0 -> cos atan inv _0") (var x complex) (assume (nonzero (cosh (asinh x)))) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (abs (tanh (asinh x))) (cos (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_log_inv_pi_I']")) (witness false (x (+ 1 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-151-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \arccos\left(\cos\left(\arcsin x\right)\right) \overset{?}{=} \left|\arcsin x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = 2\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2\)
SimpliPy acj-4 (MIT) rules_real.json[151]: acos cos asin _0 -> abs asin _0
Lean proof: included
(case sp-acj4-151-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[151]: acos cos asin _0 -> abs asin _0") (var x complex) (ask (= (acos (cos (asin x))) (abs (asin x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x 2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-153-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \arccos\left(\cos\left(\cos x\right)\right) \overset{?}{=} \left|\cos x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[153]: acos cos cos _0 -> abs cos _0
Lean proof: included
(case sp-acj4-153-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[153]: acos cos cos _0 -> abs cos _0") (var x complex) (ask (= (acos (cos (cos x))) (abs (cos x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-154-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \arccos\left(\cos\left(\sin x\right)\right) \overset{?}{=} \left|\sin x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[154]: acos cos sin _0 -> abs sin _0
Lean proof: included
(case sp-acj4-154-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[154]: acos cos sin _0 -> abs sin _0") (var x complex) (ask (= (acos (cos (sin x))) (abs (sin x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-165-complex
Given
\(x \in \mathbb{C},\quad x \neq i,\quad x \neq -i,\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \arcsin\left(\cos\left(\arctan x\right)\right) \overset{?}{=} \left|\arctan\left(\frac{1}{x}\right)\right|\)
Expected
contingent true at \(x = 1\); false at \(x = 1 + i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 1 + i\)
SimpliPy acj-4 (MIT) rules_real.json[165]: asin cos atan _0 -> abs atan inv _0
Lean proof: included
(case sp-acj4-165-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[165]: asin cos atan _0 -> abs atan inv _0") (var x complex) (assume (!= x i)) (assume (!= x (- i))) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (asin (cos (atan x))) (abs (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (+ 1 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-173-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \arcsin\left(\tanh\left(\left|x\right|\right)\right) \overset{?}{=} \arccos\left(\frac{1}{\cosh x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[173]: asin tanh abs _0 -> acos inv cosh _0
Lean proof: included
(case sp-acj4-173-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[173]: asin tanh abs _0 -> acos inv cosh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (asin (tanh (abs x))) (acos (/ (cosh x))))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-176-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \arctan\left(\left|\sinh x\right|\right) \overset{?}{=} \arccos\left(\frac{1}{\cosh x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[176]: atan abs sinh _0 -> acos inv cosh _0
Lean proof: included
(case sp-acj4-176-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[176]: atan abs sinh _0 -> acos inv cosh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (atan (abs (sinh x))) (acos (/ (cosh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-177-complex
Given
\(x \in \mathbb{C},\quad \cos x \neq 0\)
Claim
\(\displaystyle \arctan\left(\left|\tan x\right|\right) \overset{?}{=} \left|\arcsin\left(\sin x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[177]: atan abs tan _0 -> abs asin sin _0
Lean proof: included
(case sp-acj4-177-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[177]: atan abs tan _0 -> abs asin sin _0") (var x complex) (assume (nonzero (cos x))) (ask (= (atan (abs (tan x))) (abs (asin (sin x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-180-complex
Given
\(x \in \mathbb{C},\quad \sinh\left(\operatorname{arcosh}\left(x\right)\right) \neq i,\quad \sinh\left(\operatorname{arcosh}\left(x\right)\right) \neq -i,\quad \cosh\left(\operatorname{arcosh}\left(x\right)\right) \neq 0\)
Claim
\(\displaystyle \arctan\left(\sinh\left(\operatorname{arcosh}\left(x\right)\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\operatorname{arcosh}\left(x\right)\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[180]: atan sinh acosh _0 -> asin tanh acosh _0
Lean proof: included
(case sp-acj4-180-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[180]: atan sinh acosh _0 -> asin tanh acosh _0") (var x complex) (assume (!= (sinh (acosh x)) i)) (assume (!= (sinh (acosh x)) (- i))) (assume (nonzero (cosh (acosh x)))) (ask (= (atan (sinh (acosh x))) (asin (tanh (acosh x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234]; exact AE.w1_c180")))
sp-acj4-183-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \sinh\left(\log x\right) \neq i,\quad \sinh\left(\log x\right) \neq -i,\quad \cosh\left(\log x\right) \neq 0\)
Claim
\(\displaystyle \arctan\left(\sinh\left(\log x\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\log x\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[183]: atan sinh log _0 -> asin tanh log _0
Lean proof: included
(case sp-acj4-183-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[183]: atan sinh log _0 -> asin tanh log _0") (var x complex) (assume (nonzero x)) (assume (!= (sinh (log x)) i)) (assume (!= (sinh (log x)) (- i))) (assume (nonzero (cosh (log x)))) (ask (= (atan (sinh (log x))) (asin (tanh (log x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234]; exact AE.w1_c183")))
sp-acj4-189-complex
Given
\(x \in \mathbb{C},\quad \cos\left(\arcsin x\right) \neq 1,\quad \cos\left(\arcsin x\right) \neq -1,\quad x \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\cos\left(\arcsin x\right)\right) \overset{?}{=} \operatorname{arcosh}\left(\left|\frac{1}{x}\right|\right)\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[189]: atanh cos asin _0 -> acosh abs inv _0
Lean proof: included
(case sp-acj4-189-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[189]: atanh cos asin _0 -> acosh abs inv _0") (var x complex) (assume (!= (cos (asin x)) 1)) (assume (!= (cos (asin x)) -1)) (assume (nonzero x)) (ask (= (atanh (cos (asin x))) (acosh (abs (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-190-complex
Given
\(x \in \mathbb{C},\quad x \neq i,\quad x \neq -i,\quad \cos\left(\arctan x\right) \neq 1,\quad \cos\left(\arctan x\right) \neq -1,\quad x \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\cos\left(\arctan x\right)\right) \overset{?}{=} \left|\operatorname{arsinh}\left(\frac{1}{x}\right)\right|\)
Expected
contingent true at \(x = 1\); false at \(x = \frac{1}{2} \, i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = \frac{1}{2} \, i\)
SimpliPy acj-4 (MIT) rules_real.json[190]: atanh cos atan _0 -> abs asinh inv _0
Lean proof: included
(case sp-acj4-190-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[190]: atanh cos atan _0 -> abs asinh inv _0") (var x complex) (assume (!= x i)) (assume (!= x (- i))) (assume (!= (cos (atan x)) 1)) (assume (!= (cos (atan x)) -1)) (assume (nonzero x)) (ask (= (atanh (cos (atan x))) (abs (asinh (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 1/2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-191-complex
Given
\(x \in \mathbb{C},\quad \sin\left(\arccos x\right) \neq 1,\quad \sin\left(\arccos x\right) \neq -1,\quad x \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\sin\left(\arccos x\right)\right) \overset{?}{=} \operatorname{arcosh}\left(\left|\frac{1}{x}\right|\right)\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[191]: atanh sin acos _0 -> acosh abs inv _0
Lean proof: included
(case sp-acj4-191-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[191]: atanh sin acos _0 -> acosh abs inv _0") (var x complex) (assume (!= (sin (acos x)) 1)) (assume (!= (sin (acos x)) -1)) (assume (nonzero x)) (ask (= (atanh (sin (acos x))) (acosh (abs (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-20-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \sinh\left(\left|x\right|\right) \overset{?}{=} \left|\sinh x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[20] + rules.json[15]: sinh abs _0 -> abs sinh _0
Lean proof: included
(case sp-acj4-20-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[20] + rules.json[15]: sinh abs _0 -> abs sinh _0") (var x complex) (ask (= (sinh (abs x)) (abs (sinh x)))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-203-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0\)
Claim
\(\displaystyle \cos\left(\log\left(\frac{1}{x}\right)\right) \overset{?}{=} \cos\left(\log x\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[203]: cos log inv _0 -> cos log _0
Lean proof: included
(case sp-acj4-203-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[203]: cos log inv _0 -> cos log _0") (var x complex) (assume (nonzero x)) (assume (nonzero (/ x))) (ask (= (cos (log (/ x))) (cos (log x)))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-213-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{-10}{x} \neq 0\)
Claim
\(\displaystyle \frac{1}{\frac{-10}{x}} = \frac{x}{-10}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[213] + rules.json[145]: inv / (-10) !0 -> / !0 (-10)
Lean proof: included
(case sp-acj4-213-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[213] + rules.json[145]: inv / (-10) !0 -> / !0 (-10)") (var x complex) (assume (nonzero x)) (assume (nonzero (/ -10 x))) (ask (= (/ (/ -10 x)) (/ x -10))) (expect true) (lean-proof "simp_all"))
sp-acj4-216-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{-4}{x} \neq 0\)
Claim
\(\displaystyle \frac{1}{\frac{-4}{x}} = \frac{x}{-4}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[216] + rules.json[148]: inv / (-4) !0 -> / !0 (-4)
Lean proof: included
(case sp-acj4-216-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[216] + rules.json[148]: inv / (-4) !0 -> / !0 (-4)") (var x complex) (assume (nonzero x)) (assume (nonzero (/ -4 x))) (ask (= (/ (/ -4 x)) (/ x -4))) (expect true) (lean-proof "simp_all"))
sp-acj4-22-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \tanh\left(\left|x\right|\right) \overset{?}{=} \left|\tanh x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0 -> abs tanh _0
Lean proof: included
(case sp-acj4-22-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0 -> abs tanh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (tanh (abs x)) (abs (tanh x)))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-234-complex
Given
\(x \in \mathbb{C},\quad x \neq 0 \lor \operatorname{Re}\left(x\right) > 0,\quad x^{x} \neq 0,\quad x \neq 0,\quad \frac{1}{x} \neq 0 \lor \operatorname{Re}\left(x\right) > 0\)
Claim
\(\displaystyle \frac{1}{x^{x}} \overset{?}{=} \frac{1}{x}^{x}\)
Expected
contingent true at \(x = 1\); false at \(x = -\frac{1}{2}\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -\frac{1}{2}\)
SimpliPy acj-4 (MIT) rules_real.json[234]: inv pow _0 _0 -> pow inv _0 _0
Lean proof: included
(case sp-acj4-234-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[234]: inv pow _0 _0 -> pow inv _0 _0") (var x complex) (assume (or (nonzero x) (> (re x) 0))) (assume (nonzero (pow x x))) (assume (nonzero x)) (assume (or (nonzero (/ x)) (> (re x) 0))) (ask (= (/ (pow x x)) (pow (/ x) x))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1/2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-252-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle -\left|\sinh x\right| \overset{?}{=} \sinh\left(-\left|x\right|\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0 -> sinh neg abs _0
Lean proof: included
(case sp-acj4-252-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0 -> sinh neg abs _0") (var x complex) (ask (= (- (abs (sinh x))) (sinh (- (abs x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-253-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle -\left|\tanh x\right| \overset{?}{=} \tanh\left(-\left|x\right|\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0 -> tanh neg abs _0
Lean proof: included
(case sp-acj4-253-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0 -> tanh neg abs _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (- (abs (tanh x))) (tanh (- (abs x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-260-complex
Given
\(x \in \mathbb{C},\quad \cos x \neq 0,\quad \frac{1}{\cos x} \neq 0\)
Claim
\(\displaystyle -\log\left(\cos x\right) \overset{?}{=} \log\left(\frac{1}{\cos x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2\)
SimpliPy acj-4 (MIT) rules_real.json[260]: neg log cos _0 -> log inv cos _0
Lean proof: included
(case sp-acj4-260-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[260]: neg log cos _0 -> log inv cos _0") (var x complex) (assume (nonzero (cos x))) (assume (nonzero (/ (cos x)))) (ask (= (- (log (cos x))) (log (/ (cos x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x 2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-261-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \frac{1}{\cosh x} \neq 0\)
Claim
\(\displaystyle -\log\left(\cosh x\right) \overset{?}{=} \log\left(\frac{1}{\cosh x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[261]: neg log cosh _0 -> log inv cosh _0
Lean proof: included
(case sp-acj4-261-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[261]: neg log cosh _0 -> log inv cosh _0") (var x complex) (assume (nonzero (cosh x))) (assume (nonzero (/ (cosh x)))) (ask (= (- (log (cosh x))) (log (/ (cosh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-266-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0\)
Claim
\(\displaystyle -\sin\left(\log x\right) \overset{?}{=} \sin\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[266]: neg sin log _0 -> sin log inv _0
Lean proof: included
(case sp-acj4-266-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[266]: neg sin log _0 -> sin log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (/ x))) (ask (= (- (sin (log x))) (sin (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-267-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \cos\left(\log x\right) \neq 0,\quad \frac{1}{x} \neq 0,\quad \cos\left(\log\left(\frac{1}{x}\right)\right) \neq 0\)
Claim
\(\displaystyle -\tan\left(\log x\right) \overset{?}{=} \tan\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[267]: neg tan log _0 -> tan log inv _0
Lean proof: included
(case sp-acj4-267-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[267]: neg tan log _0 -> tan log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (cos (log x)))) (assume (nonzero (/ x))) (assume (nonzero (cos (log (/ x))))) (ask (= (- (tan (log x))) (tan (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-274-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \left|x\right|^{-2} \overset{?}{=} x^{-2}\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[274] + rules.json[183]: pow abs _0 (-2) -> pow _0 (-2)
Lean proof: included
(case sp-acj4-274-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[274] + rules.json[183]: pow abs _0 (-2) -> pow _0 (-2)") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (pow (abs x) -2) (pow x -2))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-278-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \left|x\right|^{-6} \overset{?}{=} x^{-6}\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[278] + rules.json[187]: pow abs _0 (-6) -> pow _0 (-6)
Lean proof: included
(case sp-acj4-278-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[278] + rules.json[187]: pow abs _0 (-6) -> pow _0 (-6)") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (pow (abs x) -6) (pow x -6))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-283-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right|^{2} \overset{?}{=} x \, x\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2 -> * _0 _0
Lean proof: included
(case sp-acj4-283-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2 -> * _0 _0") (var x complex) (ask (= (pow (abs x) 2) (* x x))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")))
sp-acj4-284-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right|^{3} = \left|x^{3}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[284] + rules.json[193]: pow abs _0 3 -> abs pow _0 3
Lean proof: included
(case sp-acj4-284-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[284] + rules.json[193]: pow abs _0 3 -> abs pow _0 3") (var x complex) (ask (= (pow (abs x) 3) (abs (pow x 3)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-288-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right|^{7} = \left|x^{7}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[288] + rules.json[197]: pow abs _0 7 -> abs pow _0 7
Lean proof: included
(case sp-acj4-288-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[288] + rules.json[197]: pow abs _0 7 -> abs pow _0 7") (var x complex) (ask (= (pow (abs x) 7) (abs (pow x 7)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-30-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle 4 \, \left|x\right| = \left|-4 \, x\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[30] + rules.json[23]: * 4 abs _0 -> abs * (-4) _0
Lean proof: included
(case sp-acj4-30-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[30] + rules.json[23]: * 4 abs _0 -> abs * (-4) _0") (var x complex) (ask (= (* 4 (abs x)) (abs (* -4 x)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-334-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0 \lor \operatorname{Re}\left(e^{1}\right) > 0\)
Claim
\(\displaystyle \left(e^{x}\right)^{e^{1}} \overset{?}{=} e^{e^{1} \, x}\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[334]: pow exp _0 np.e -> exp * np.e _0
Lean proof: included
(case sp-acj4-334-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[334]: pow exp _0 np.e -> exp * np.e _0") (var x complex) (assume (or (nonzero (exp x)) (> (re (exp 1)) 0))) (ask (= (pow (exp x) (exp 1)) (exp (* (exp 1) x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-335-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0\)
Claim
\(\displaystyle \left(e^{x}\right)^{\pi} \overset{?}{=} e^{\pi \, x}\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[335]: pow exp _0 np.pi -> exp * np.pi _0
Lean proof: included
(case sp-acj4-335-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[335]: pow exp _0 np.pi -> exp * np.pi _0") (var x complex) (assume (or (nonzero (exp x)) (> (re pi) 0))) (ask (= (pow (exp x) pi) (exp (* pi x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-336-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0 \lor \operatorname{Re}\left(x\right) > 0\)
Claim
\(\displaystyle \left(e^{x}\right)^{x} \overset{?}{=} e^{x \, x}\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[336]: pow exp _0 _0 -> exp * _0 _0
Lean proof: included
(case sp-acj4-336-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[336]: pow exp _0 _0 -> exp * _0 _0") (var x complex) (assume (or (nonzero (exp x)) (> (re x) 0))) (ask (= (pow (exp x) x) (exp (* x x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-343-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \frac{1}{\pi}^{x} = \pi^{-x}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[343]: pow inv np.pi _0 -> pow np.pi neg _0
Lean proof: included
(case sp-acj4-343-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[343]: pow inv np.pi _0 -> pow np.pi neg _0") (var x complex) (ask (= (pow (/ pi) x) (pow pi (- x)))) (expect true) (lean-proof "refine ⟨by simp [Real.pi_ne_zero], ?_⟩; simp only [AE.pi_def, AE.inv_def]; exact AE.pow_inv_base Real.pi_pos x"))
sp-acj4-344-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0 \lor \operatorname{Re}\left(x\right) > 0,\quad x \neq 0 \lor \operatorname{Re}\left(-x\right) > 0\)
Claim
\(\displaystyle \frac{1}{x}^{x} \overset{?}{=} x^{-x}\)
Expected
contingent true at \(x = 1\); false at \(x = -\frac{1}{2}\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -\frac{1}{2}\)
SimpliPy acj-4 (MIT) rules_real.json[344]: pow inv _0 _0 -> pow _0 neg _0
Lean proof: included
(case sp-acj4-344-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[344]: pow inv _0 _0 -> pow _0 neg _0") (var x complex) (assume (nonzero x)) (assume (or (nonzero (/ x)) (> (re x) 0))) (assume (or (nonzero x) (> (re (- x)) 0))) (ask (= (pow (/ x) x) (pow x (- x)))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1/2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-368-complex
Given
\(x \in \mathbb{C},\quad \arccos x \in \mathbb{R},\quad \arccos x \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(\arccos x, -5\right) = \left(\arccos x\right)^{-\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5) -> pow acos _0 -0.2
Lean proof: included
(case sp-acj4-368-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5) -> pow acos _0 -0.2") (var x complex) (assume (real (acos x))) (assume (nonzero (acos x))) (ask (= (surd (acos x) -5) (pow (acos x) -1/5))) (expect true) (lean-proof "have hi : (AE.acos x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acos_re_nonneg hi; refine ⟨⟨⟨AE.acos.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨-5, by norm_num⟩ : ∃ n : ℤ, (-5:ℂ) = n), by norm_num, Or.inl hr, Or.inl h1.2⟩⟩, AE.acos.D_true x, (AE.pow.D_iff _ _).2 (Or.inl h1.2)⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]; norm_num"))
sp-acj4-369-complex
Given
\(x \in \mathbb{C},\quad \arccos x \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(\arccos x, 5\right) = \left(\arccos x\right)^{\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5 -> pow acos _0 0.2
Lean proof: included
(case sp-acj4-369-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5 -> pow acos _0 0.2") (var x complex) (assume (real (acos x))) (ask (= (surd (acos x) 5) (pow (acos x) 1/5))) (expect true) (lean-proof "have hi : (AE.acos x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acos_re_nonneg hi; refine ⟨⟨⟨AE.acos.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨5, by norm_num⟩ : ∃ n : ℤ, (5:ℂ) = n), by norm_num, Or.inl hr, Or.inr (by norm_num)⟩⟩, AE.acos.D_true x, (AE.pow.D_iff _ _).2 (Or.inr (by norm_num))⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]"))
sp-acj4-370-complex
Given
\(x \in \mathbb{C},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), -5\right) = \operatorname{arcosh}\left(x\right)^{-\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _0 -0.2
Lean proof: included
(case sp-acj4-370-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _0 -0.2") (var x complex) (assume (real (acosh x))) (assume (nonzero (acosh x))) (ask (= (surd (acosh x) -5) (pow (acosh x) -1/5))) (expect true) (lean-proof "have hi : (AE.acosh x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acosh_re_nonneg hi; refine ⟨⟨⟨AE.acosh.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨-5, by norm_num⟩ : ∃ n : ℤ, (-5:ℂ) = n), by norm_num, Or.inl hr, Or.inl h1.2⟩⟩, AE.acosh.D_true x, (AE.pow.D_iff _ _).2 (Or.inl h1.2)⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]; norm_num"))
sp-acj4-371-complex
Given
\(x \in \mathbb{C},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), 5\right) = \operatorname{arcosh}\left(x\right)^{\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5 -> pow acosh _0 0.2
Lean proof: included
(case sp-acj4-371-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5 -> pow acosh _0 0.2") (var x complex) (assume (real (acosh x))) (ask (= (surd (acosh x) 5) (pow (acosh x) 1/5))) (expect true) (lean-proof "have hi : (AE.acosh x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acosh_re_nonneg hi; refine ⟨⟨⟨AE.acosh.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨5, by norm_num⟩ : ∃ n : ℤ, (5:ℂ) = n), by norm_num, Or.inl hr, Or.inr (by norm_num)⟩⟩, AE.acosh.D_true x, (AE.pow.D_iff _ _).2 (Or.inr (by norm_num))⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]"))
sp-acj4-372-complex
Given
\(x \in \mathbb{C},\quad \cosh x \in \mathbb{R},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(\cosh x, -5\right) \overset{?}{=} \left(\cosh x\right)^{-\frac{1}{5}}\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5) -> pow cosh _0 -0.2
Lean proof: included
(case sp-acj4-372-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5) -> pow cosh _0 -0.2") (var x complex) (assume (real (cosh x))) (assume (nonzero (cosh x))) (ask (= (surd (cosh x) -5) (pow (cosh x) -1/5))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-373-complex
Given
\(x \in \mathbb{C},\quad \cosh x \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(\cosh x, 5\right) \overset{?}{=} \left(\cosh x\right)^{\frac{1}{5}}\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5 -> pow cosh _0 0.2
Lean proof: included
(case sp-acj4-373-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5 -> pow cosh _0 0.2") (var x complex) (assume (real (cosh x))) (ask (= (surd (cosh x) 5) (pow (cosh x) 1/5))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-387-complex
Given
\(x \in \mathbb{C},\quad e^{x} \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(e^{x}, 5\right) \overset{?}{=} e^{\frac{x}{5}}\)
Expected
contingent true at \(x = 0\); false at \(x = \pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = \pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[387]: rootn exp _0 5 -> exp / _0 5
Lean proof: included
(case sp-acj4-387-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[387]: rootn exp _0 5 -> exp / _0 5") (var x complex) (assume (real (exp x))) (ask (= (surd (exp x) 5) (exp (/ x 5)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-389-complex
Given
\(x \in \mathbb{C},\quad e^{x} \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(e^{x}, 7\right) \overset{?}{=} e^{\frac{x}{7}}\)
Expected
contingent true at \(x = 0\); false at \(x = \pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = \pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[389]: rootn exp _0 7 -> exp / _0 7
Lean proof: included
(case sp-acj4-389-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[389]: rootn exp _0 7 -> exp / _0 7") (var x complex) (assume (real (exp x))) (ask (= (surd (exp x) 7) (exp (/ x 7)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-396-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \sin\left(\left|\sin x\right|\right) \overset{?}{=} \left|\sin\left(\sin x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0 -> abs sin sin _0
Lean proof: included
(case sp-acj4-396-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0 -> abs sin sin _0") (var x complex) (ask (= (sin (abs (sin x))) (abs (sin (sin x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-40-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right| \, \pi = \left|\pi \, x\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[40] + rules.json[33]: * abs _0 np.pi -> abs * np.pi _0
Lean proof: included
(case sp-acj4-40-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[40] + rules.json[33]: * abs _0 np.pi -> abs * np.pi _0") (var x complex) (ask (= (* (abs x) pi) (abs (* pi x)))) (expect true) (lean-proof "simp [AE.abs_eq]; ring"))
sp-acj4-403-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0\)
Claim
\(\displaystyle \sin\left(-\log x\right) \overset{?}{=} \sin\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[403]: sin neg log _0 -> sin log inv _0
Lean proof: included
(case sp-acj4-403-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[403]: sin neg log _0 -> sin log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (/ x))) (ask (= (sin (- (log x))) (sin (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-404-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \sin\left(\tanh\left(\left|x\right|\right)\right) \overset{?}{=} \left|\sin\left(\tanh x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0 -> abs sin tanh _0
Lean proof: included
(case sp-acj4-404-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0 -> abs sin tanh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (sin (tanh (abs x))) (abs (sin (tanh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-414-complex
Given
\(x \in \mathbb{C},\quad \cos\left(\left|\sin x\right|\right) \neq 0,\quad \cos\left(\sin x\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\left|\sin x\right|\right) \overset{?}{=} \left|\tan\left(\sin x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0 -> abs tan sin _0
Lean proof: included
(case sp-acj4-414-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0 -> abs tan sin _0") (var x complex) (assume (nonzero (cos (abs (sin x))))) (assume (nonzero (cos (sin x)))) (ask (= (tan (abs (sin x))) (abs (tan (sin x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-415-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \cos\left(\left|\tanh x\right|\right) \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\left|\tanh x\right|\right) \overset{?}{=} \left|\tan\left(\tanh x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0 -> abs tan tanh _0
Lean proof: included
(case sp-acj4-415-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0 -> abs tan tanh _0") (var x complex) (assume (nonzero (cosh x))) (assume (nonzero (cos (abs (tanh x))))) (assume (nonzero (cos (tanh x)))) (ask (= (tan (abs (tanh x))) (abs (tan (tanh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-420-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \cos\left(-\log x\right) \neq 0,\quad \frac{1}{x} \neq 0,\quad \cos\left(\log\left(\frac{1}{x}\right)\right) \neq 0\)
Claim
\(\displaystyle \tan\left(-\log x\right) \overset{?}{=} \tan\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[420]: tan neg log _0 -> tan log inv _0
Lean proof: included
(case sp-acj4-420-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[420]: tan neg log _0 -> tan log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (cos (- (log x))))) (assume (nonzero (/ x))) (assume (nonzero (cos (log (/ x))))) (ask (= (tan (- (log x))) (tan (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-421-complex
Given
\(x \in \mathbb{C},\quad \cos\left(\tanh\left(\left|x\right|\right)\right) \neq 0,\quad \cosh x \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\tanh\left(\left|x\right|\right)\right) \overset{?}{=} \left|\tan\left(\tanh x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0 -> abs tan tanh _0
Lean proof: included
(case sp-acj4-421-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0 -> abs tan tanh _0") (var x complex) (assume (nonzero (cos (tanh (abs x))))) (assume (nonzero (cosh x))) (assume (nonzero (cos (tanh x)))) (ask (= (tan (tanh (abs x))) (abs (tan (tanh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-422-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \tanh\left(\left|\operatorname{arsinh}\left(x\right)\right|\right) \overset{?}{=} \cos\left(\arctan\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[422]: tanh abs asinh _0 -> cos atan inv _0
Lean proof: included
(case sp-acj4-422-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[422]: tanh abs asinh _0 -> cos atan inv _0") (var x complex) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (tanh (abs (asinh x))) (cos (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-423-complex
Given
\(x \in \mathbb{C},\quad \cosh\left(\operatorname{arcosh}\left(\left|x\right|\right)\right) \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \tanh\left(\operatorname{arcosh}\left(\left|x\right|\right)\right) \overset{?}{=} \cos\left(\arcsin\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[423]: tanh acosh abs _0 -> cos asin inv _0
Lean proof: included
(case sp-acj4-423-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[423]: tanh acosh abs _0 -> cos asin inv _0") (var x complex) (assume (nonzero (cosh (acosh (abs x))))) (assume (nonzero x)) (ask (= (tanh (acosh (abs x))) (cos (asin (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-424-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \tanh\left(\operatorname{arsinh}\left(\left|x\right|\right)\right) \overset{?}{=} \cos\left(\arctan\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[424]: tanh asinh abs _0 -> cos atan inv _0
Lean proof: included
(case sp-acj4-424-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[424]: tanh asinh abs _0 -> cos atan inv _0") (var x complex) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (tanh (asinh (abs x))) (cos (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-430-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(x\right) > 0,\quad x^{x} \neq 0\)
Claim
\(\displaystyle \log x \, x \overset{?}{=} \log\left(x^{x}\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[430]: * log ?0 ?0 -> log pow ?0 ?0
Lean proof: included
(case sp-acj4-430-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[430]: * log ?0 ?0 -> log pow ?0 ?0") (var x complex) (assume (nonzero x)) (assume (or (nonzero x) (> (re x) 0))) (assume (nonzero (pow x x))) (ask (= (* (log x) x) (log (pow x x)))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-433-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad -\log x \neq i,\quad -\log x \neq -i,\quad \frac{1}{x} \neq 0,\quad \log\left(\frac{1}{x}\right) \neq i,\quad \log\left(\frac{1}{x}\right) \neq -i\)
Claim
\(\displaystyle \arctan\left(-\log x\right) \overset{?}{=} \arctan\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[433]: atan neg log !0 -> atan log inv !0
Lean proof: included
(case sp-acj4-433-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[433]: atan neg log !0 -> atan log inv !0") (var x complex) (assume (nonzero x)) (assume (!= (- (log x)) i)) (assume (!= (- (log x)) (- i))) (assume (nonzero (/ x))) (assume (!= (log (/ x)) i)) (assume (!= (log (/ x)) (- i))) (ask (= (atan (- (log x))) (atan (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234]; exact AE.w1_c433")))
sp-acj4-442-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \log x \neq 0,\quad \frac{1}{\log x} \neq 0\)
Claim
\(\displaystyle -\log\left(\log x\right) \overset{?}{=} \log\left(\frac{1}{\log x}\right)\)
Expected
contingent true at \(x = -1\); false at \(x = \frac{1}{2}\)
equal at some admissible points, different at others
witness (true): \(x = -1\)
witness (false): \(x = \frac{1}{2}\)
SimpliPy acj-4 (MIT) rules_real.json[442]: neg log log ?0 -> log inv log ?0
Lean proof: included
(case sp-acj4-442-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[442]: neg log log ?0 -> log inv log ?0") (var x complex) (assume (nonzero x)) (assume (nonzero (log x))) (assume (nonzero (/ (log x)))) (ask (= (- (log (log x))) (log (/ (log x))))) (expect contingent) (witness true (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x 1/2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-446-complex
Given
\(x \in \mathbb{C},\quad x \neq i,\quad x \neq -i,\quad \cos\left(\left|\arctan x\right|\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\left|\arctan x\right|\right) \overset{?}{=} \left|x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = \frac{1}{2} \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = \frac{1}{2} \, i\)
SimpliPy acj-4 (MIT) rules_real.json[446]: tan abs atan !0 -> abs !0
Lean proof: included
(case sp-acj4-446-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[446]: tan abs atan !0 -> abs !0") (var x complex) (assume (!= x i)) (assume (!= x (- i))) (assume (nonzero (cos (abs (atan x))))) (ask (= (tan (abs (atan x))) (abs x))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x (* 1/2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-450-complex
Given
\(x \in \mathbb{C},\quad x \neq 0\)
Claim
\(\displaystyle \frac{x}{x} = 1\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[450]: / $0 $0 -> 1
Lean proof: included
(case sp-acj4-450-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[450]: / $0 $0 -> 1") (var x complex) (assume (nonzero x)) (ask (= (/ x x) 1)) (expect true) (lean-proof "simp_all"))
sp-acj4-451-complex
Given
\(x \in \mathbb{C},\quad x \neq 0\)
Claim
\(\displaystyle \frac{0}{x} = 0\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[451]: / 0 $0 -> 0
Lean proof: included
(case sp-acj4-451-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[451]: / 0 $0 -> 0") (var x complex) (assume (nonzero x)) (ask (= (/ 0 x) 0)) (expect true) (lean-proof "simp_all"))
sp-acj4-50-complex
Given
\(x \in \mathbb{C},\quad 6^{x} \neq 0\)
Claim
\(\displaystyle \log 6 \, x \overset{?}{=} \log\left(6^{x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[50]: * log 6 _0 -> log pow 6 _0
Lean proof: included
(case sp-acj4-50-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[50]: * log 6 _0 -> log pow 6 _0") (var x complex) (assume (nonzero (pow 6 x))) (ask (= (* (log 6) x) (log (pow 6 x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-51-complex
Given
\(x \in \mathbb{C},\quad 7^{x} \neq 0\)
Claim
\(\displaystyle \log 7 \, x \overset{?}{=} \log\left(7^{x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[51]: * log 7 _0 -> log pow 7 _0
Lean proof: included
(case sp-acj4-51-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[51]: * log 7 _0 -> log pow 7 _0") (var x complex) (assume (nonzero (pow 7 x))) (ask (= (* (log 7) x) (log (pow 7 x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-55-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(e^{1}\right) > 0,\quad x^{e^{1}} \neq 0\)
Claim
\(\displaystyle \log x \, e^{1} \overset{?}{=} \log\left(x^{e^{1}}\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[55]: * log _0 np.e -> log pow _0 np.e
Lean proof: included
(case sp-acj4-55-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[55]: * log _0 np.e -> log pow _0 np.e") (var x complex) (assume (nonzero x)) (assume (or (nonzero x) (> (re (exp 1)) 0))) (assume (nonzero (pow x (exp 1)))) (ask (= (* (log x) (exp 1)) (log (pow x (exp 1))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-56-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0,\quad x^{\pi} \neq 0\)
Claim
\(\displaystyle \log x \, \pi \overset{?}{=} \log\left(x^{\pi}\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[56]: * log _0 np.pi -> log pow _0 np.pi
Lean proof: included
(case sp-acj4-56-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[56]: * log _0 np.pi -> log pow _0 np.pi") (var x complex) (assume (nonzero x)) (assume (or (nonzero x) (> (re pi) 0))) (assume (nonzero (pow x pi))) (ask (= (* (log x) pi) (log (pow x pi)))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-6-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \tanh x \neq 1,\quad \tanh x \neq -1\)
Claim
\(\displaystyle \operatorname{artanh}\left(\tanh x\right) \overset{?}{=} x\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[6]: atanh tanh !0 -> !0
Lean proof: included
(case sp-acj4-6-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[6]: atanh tanh !0 -> !0") (var x complex) (assume (nonzero (cosh x))) (assume (!= (tanh x) 1)) (assume (!= (tanh x) -1)) (ask (= (atanh (tanh x)) x)) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-7-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{1}{\left|x\right|} = \left|\frac{1}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[7] + rules.json[4]: inv abs _0 -> abs inv _0
Lean proof: included
(case sp-acj4-7-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[7] + rules.json[4]: inv abs _0 -> abs inv _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ (abs x)) (abs (/ x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-9-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0\)
Claim
\(\displaystyle \log\left(e^{x}\right) \overset{?}{=} x\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[9]: log exp _0 -> _0
Lean proof: included
(case sp-acj4-9-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[9]: log exp _0 -> _0") (var x complex) (assume (nonzero (exp x))) (ask (= (log (exp x)) x)) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-90-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{3}{\left|x\right|} = \left|\frac{-3}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0 -> abs / (-3) _0
Lean proof: included
(case sp-acj4-90-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0 -> abs / (-3) _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ 3 (abs x)) (abs (/ -3 x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-94-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{7}{\left|x\right|} = \left|\frac{-7}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[94] + rules.json[73]: / 7 abs _0 -> abs / (-7) _0
Lean proof: included
(case sp-acj4-94-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[94] + rules.json[73]: / 7 abs _0 -> abs / (-7) _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ 7 (abs x)) (abs (/ -7 x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-99-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \frac{\left|x\right|}{3} = \left|\frac{x}{-3}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3 -> abs / _0 (-3)
Lean proof: included
(case sp-acj4-99-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3 -> abs / _0 (-3)") (var x complex) (ask (= (/ (abs x) 3) (abs (/ x -3)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
Case set ee702ae2fe06
sp-acj4-0-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \operatorname{arcosh}\left(\cosh x\right) \overset{?}{=} \left|x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[0]: acosh cosh _0 -> abs _0
Lean proof: included
(case sp-acj4-0-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[0]: acosh cosh _0 -> abs _0") (var x complex) (ask (= (acosh (cosh x)) (abs x))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-102-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \frac{\left|x\right|}{6} = \left|\frac{x}{-6}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6 -> abs / _0 (-6)
Lean proof: included
(case sp-acj4-102-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6 -> abs / _0 (-6)") (var x complex) (ask (= (/ (abs x) 6) (abs (/ x -6)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-108-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \left|x\right| \neq 0\)
Claim
\(\displaystyle \frac{\left|x\right|}{x} \overset{?}{=} \frac{x}{\left|x\right|}\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0 -> / _0 abs _0
Lean proof: included
(case sp-acj4-108-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0 -> / _0 abs _0") (var x complex) (assume (nonzero x)) (assume (nonzero (abs x))) (ask (= (/ (abs x) x) (/ x (abs x)))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")))
sp-acj4-120-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{\pi}{\left|x\right|} = \left|\frac{\pi}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0 -> abs / np.pi _0
Lean proof: included
(case sp-acj4-120-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0 -> abs / np.pi _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ pi (abs x)) (abs (/ pi x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-130-complex
Given
\(x \in \mathbb{C},\quad \cos x \neq 0,\quad \frac{1}{\cos x} \neq 0\)
Claim
\(\displaystyle \left|\log\left(\cos x\right)\right| \overset{?}{=} \log\left(\frac{1}{\cos x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2\)
SimpliPy acj-4 (MIT) rules_real.json[130]: abs log cos _0 -> log inv cos _0
Lean proof: included
(case sp-acj4-130-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[130]: abs log cos _0 -> log inv cos _0") (var x complex) (assume (nonzero (cos x))) (assume (nonzero (/ (cos x)))) (ask (= (abs (log (cos x))) (log (/ (cos x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x 2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-131-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \left|\log\left(\cosh x\right)\right| \overset{?}{=} \log\left(\cosh x\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0 -> log cosh _0
Lean proof: included
(case sp-acj4-131-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0 -> log cosh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (abs (log (cosh x))) (log (cosh x)))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-132-complex
Given
\(x \in \mathbb{C},\quad \sin x \neq 0\)
Claim
\(\displaystyle \left|\log\left(\sin x\right)\right| \overset{?}{=} -\log\left(\sin x\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0 -> neg log sin _0
Lean proof: included
(case sp-acj4-132-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0 -> neg log sin _0") (var x complex) (assume (nonzero (sin x))) (ask (= (abs (log (sin x))) (- (log (sin x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-133-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \tanh x \neq 0\)
Claim
\(\displaystyle \left|\log\left(\tanh x\right)\right| \overset{?}{=} -\log\left(\tanh x\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0 -> neg log tanh _0
Lean proof: included
(case sp-acj4-133-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0 -> neg log tanh _0") (var x complex) (assume (nonzero (cosh x))) (assume (nonzero (tanh x))) (ask (= (abs (log (tanh x))) (- (log (tanh x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-137-complex
Given
\(x \in \mathbb{C},\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0\)
Claim
\(\displaystyle \left|x^{\pi}\right| \overset{?}{=} x^{\pi}\)
Expected
contingent true at \(x = 0\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi -> pow _0 np.pi
Lean proof: included
(case sp-acj4-137-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi -> pow _0 np.pi") (var x complex) (assume (or (nonzero x) (> (re pi) 0))) (ask (= (abs (pow x pi)) (pow x pi))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-147-complex
Given
\(x \in \mathbb{C},\quad \cosh\left(\operatorname{arsinh}\left(x\right)\right) \neq 0,\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \left|\tanh\left(\operatorname{arsinh}\left(x\right)\right)\right| \overset{?}{=} \cos\left(\arctan\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = 1 + i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 1 + i\)
SimpliPy acj-4 (MIT) rules_real.json[147]: abs tanh asinh _0 -> cos atan inv _0
Lean proof: included
(case sp-acj4-147-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[147]: abs tanh asinh _0 -> cos atan inv _0") (var x complex) (assume (nonzero (cosh (asinh x)))) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (abs (tanh (asinh x))) (cos (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_log_inv_pi_I']")) (witness false (x (+ 1 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-151-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \arccos\left(\cos\left(\arcsin x\right)\right) \overset{?}{=} \left|\arcsin x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = 2\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2\)
SimpliPy acj-4 (MIT) rules_real.json[151]: acos cos asin _0 -> abs asin _0
Lean proof: included
(case sp-acj4-151-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[151]: acos cos asin _0 -> abs asin _0") (var x complex) (ask (= (acos (cos (asin x))) (abs (asin x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x 2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-153-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \arccos\left(\cos\left(\cos x\right)\right) \overset{?}{=} \left|\cos x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[153]: acos cos cos _0 -> abs cos _0
Lean proof: included
(case sp-acj4-153-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[153]: acos cos cos _0 -> abs cos _0") (var x complex) (ask (= (acos (cos (cos x))) (abs (cos x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-154-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \arccos\left(\cos\left(\sin x\right)\right) \overset{?}{=} \left|\sin x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[154]: acos cos sin _0 -> abs sin _0
Lean proof: included
(case sp-acj4-154-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[154]: acos cos sin _0 -> abs sin _0") (var x complex) (ask (= (acos (cos (sin x))) (abs (sin x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-165-complex
Given
\(x \in \mathbb{C},\quad x \neq i,\quad x \neq -i,\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \arcsin\left(\cos\left(\arctan x\right)\right) \overset{?}{=} \left|\arctan\left(\frac{1}{x}\right)\right|\)
Expected
contingent true at \(x = 1\); false at \(x = 1 + i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 1 + i\)
SimpliPy acj-4 (MIT) rules_real.json[165]: asin cos atan _0 -> abs atan inv _0
Lean proof: included
(case sp-acj4-165-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[165]: asin cos atan _0 -> abs atan inv _0") (var x complex) (assume (!= x i)) (assume (!= x (- i))) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (asin (cos (atan x))) (abs (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (+ 1 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-173-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \arcsin\left(\tanh\left(\left|x\right|\right)\right) \overset{?}{=} \arccos\left(\frac{1}{\cosh x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[173]: asin tanh abs _0 -> acos inv cosh _0
Lean proof: included
(case sp-acj4-173-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[173]: asin tanh abs _0 -> acos inv cosh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (asin (tanh (abs x))) (acos (/ (cosh x))))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-176-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \arctan\left(\left|\sinh x\right|\right) \overset{?}{=} \arccos\left(\frac{1}{\cosh x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[176]: atan abs sinh _0 -> acos inv cosh _0
Lean proof: included
(case sp-acj4-176-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[176]: atan abs sinh _0 -> acos inv cosh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (atan (abs (sinh x))) (acos (/ (cosh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-177-complex
Given
\(x \in \mathbb{C},\quad \cos x \neq 0\)
Claim
\(\displaystyle \arctan\left(\left|\tan x\right|\right) \overset{?}{=} \left|\arcsin\left(\sin x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[177]: atan abs tan _0 -> abs asin sin _0
Lean proof: included
(case sp-acj4-177-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[177]: atan abs tan _0 -> abs asin sin _0") (var x complex) (assume (nonzero (cos x))) (ask (= (atan (abs (tan x))) (abs (asin (sin x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-180-complex
Given
\(x \in \mathbb{C},\quad \sinh\left(\operatorname{arcosh}\left(x\right)\right) \neq i,\quad \sinh\left(\operatorname{arcosh}\left(x\right)\right) \neq -i,\quad \cosh\left(\operatorname{arcosh}\left(x\right)\right) \neq 0\)
Claim
\(\displaystyle \arctan\left(\sinh\left(\operatorname{arcosh}\left(x\right)\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\operatorname{arcosh}\left(x\right)\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[180]: atan sinh acosh _0 -> asin tanh acosh _0
Lean proof: included
(case sp-acj4-180-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[180]: atan sinh acosh _0 -> asin tanh acosh _0") (var x complex) (assume (!= (sinh (acosh x)) i)) (assume (!= (sinh (acosh x)) (- i))) (assume (nonzero (cosh (acosh x)))) (ask (= (atan (sinh (acosh x))) (asin (tanh (acosh x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234]; exact AE.w1_c180")))
sp-acj4-183-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \sinh\left(\log x\right) \neq i,\quad \sinh\left(\log x\right) \neq -i,\quad \cosh\left(\log x\right) \neq 0\)
Claim
\(\displaystyle \arctan\left(\sinh\left(\log x\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\log x\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[183]: atan sinh log _0 -> asin tanh log _0
Lean proof: included
(case sp-acj4-183-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[183]: atan sinh log _0 -> asin tanh log _0") (var x complex) (assume (nonzero x)) (assume (!= (sinh (log x)) i)) (assume (!= (sinh (log x)) (- i))) (assume (nonzero (cosh (log x)))) (ask (= (atan (sinh (log x))) (asin (tanh (log x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234]; exact AE.w1_c183")))
sp-acj4-189-complex
Given
\(x \in \mathbb{C},\quad \cos\left(\arcsin x\right) \neq 1,\quad \cos\left(\arcsin x\right) \neq -1,\quad x \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\cos\left(\arcsin x\right)\right) \overset{?}{=} \operatorname{arcosh}\left(\left|\frac{1}{x}\right|\right)\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[189]: atanh cos asin _0 -> acosh abs inv _0
Lean proof: included
(case sp-acj4-189-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[189]: atanh cos asin _0 -> acosh abs inv _0") (var x complex) (assume (!= (cos (asin x)) 1)) (assume (!= (cos (asin x)) -1)) (assume (nonzero x)) (ask (= (atanh (cos (asin x))) (acosh (abs (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-190-complex
Given
\(x \in \mathbb{C},\quad x \neq i,\quad x \neq -i,\quad \cos\left(\arctan x\right) \neq 1,\quad \cos\left(\arctan x\right) \neq -1,\quad x \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\cos\left(\arctan x\right)\right) \overset{?}{=} \left|\operatorname{arsinh}\left(\frac{1}{x}\right)\right|\)
Expected
contingent true at \(x = 1\); false at \(x = \frac{1}{2} \, i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = \frac{1}{2} \, i\)
SimpliPy acj-4 (MIT) rules_real.json[190]: atanh cos atan _0 -> abs asinh inv _0
Lean proof: included
(case sp-acj4-190-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[190]: atanh cos atan _0 -> abs asinh inv _0") (var x complex) (assume (!= x i)) (assume (!= x (- i))) (assume (!= (cos (atan x)) 1)) (assume (!= (cos (atan x)) -1)) (assume (nonzero x)) (ask (= (atanh (cos (atan x))) (abs (asinh (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 1/2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-191-complex
Given
\(x \in \mathbb{C},\quad \sin\left(\arccos x\right) \neq 1,\quad \sin\left(\arccos x\right) \neq -1,\quad x \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\sin\left(\arccos x\right)\right) \overset{?}{=} \operatorname{arcosh}\left(\left|\frac{1}{x}\right|\right)\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[191]: atanh sin acos _0 -> acosh abs inv _0
Lean proof: included
(case sp-acj4-191-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[191]: atanh sin acos _0 -> acosh abs inv _0") (var x complex) (assume (!= (sin (acos x)) 1)) (assume (!= (sin (acos x)) -1)) (assume (nonzero x)) (ask (= (atanh (sin (acos x))) (acosh (abs (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-20-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \sinh\left(\left|x\right|\right) \overset{?}{=} \left|\sinh x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[20] + rules.json[15]: sinh abs _0 -> abs sinh _0
Lean proof: included
(case sp-acj4-20-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[20] + rules.json[15]: sinh abs _0 -> abs sinh _0") (var x complex) (ask (= (sinh (abs x)) (abs (sinh x)))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-203-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0\)
Claim
\(\displaystyle \cos\left(\log\left(\frac{1}{x}\right)\right) \overset{?}{=} \cos\left(\log x\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[203]: cos log inv _0 -> cos log _0
Lean proof: included
(case sp-acj4-203-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[203]: cos log inv _0 -> cos log _0") (var x complex) (assume (nonzero x)) (assume (nonzero (/ x))) (ask (= (cos (log (/ x))) (cos (log x)))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-213-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{-10}{x} \neq 0\)
Claim
\(\displaystyle \frac{1}{\frac{-10}{x}} = \frac{x}{-10}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[213] + rules.json[145]: inv / (-10) !0 -> / !0 (-10)
Lean proof: included
(case sp-acj4-213-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[213] + rules.json[145]: inv / (-10) !0 -> / !0 (-10)") (var x complex) (assume (nonzero x)) (assume (nonzero (/ -10 x))) (ask (= (/ (/ -10 x)) (/ x -10))) (expect true) (lean-proof "simp_all"))
sp-acj4-216-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{-4}{x} \neq 0\)
Claim
\(\displaystyle \frac{1}{\frac{-4}{x}} = \frac{x}{-4}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[216] + rules.json[148]: inv / (-4) !0 -> / !0 (-4)
Lean proof: included
(case sp-acj4-216-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[216] + rules.json[148]: inv / (-4) !0 -> / !0 (-4)") (var x complex) (assume (nonzero x)) (assume (nonzero (/ -4 x))) (ask (= (/ (/ -4 x)) (/ x -4))) (expect true) (lean-proof "simp_all"))
sp-acj4-22-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \tanh\left(\left|x\right|\right) \overset{?}{=} \left|\tanh x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0 -> abs tanh _0
Lean proof: included
(case sp-acj4-22-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0 -> abs tanh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (tanh (abs x)) (abs (tanh x)))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-234-complex
Given
\(x \in \mathbb{C},\quad x \neq 0 \lor \operatorname{Re}\left(x\right) > 0,\quad x^{x} \neq 0,\quad x \neq 0,\quad \frac{1}{x} \neq 0 \lor \operatorname{Re}\left(x\right) > 0\)
Claim
\(\displaystyle \frac{1}{x^{x}} \overset{?}{=} \frac{1}{x}^{x}\)
Expected
contingent true at \(x = 1\); false at \(x = -\frac{1}{2}\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -\frac{1}{2}\)
SimpliPy acj-4 (MIT) rules_real.json[234]: inv pow _0 _0 -> pow inv _0 _0
Lean proof: included
(case sp-acj4-234-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[234]: inv pow _0 _0 -> pow inv _0 _0") (var x complex) (assume (or (nonzero x) (> (re x) 0))) (assume (nonzero (pow x x))) (assume (nonzero x)) (assume (or (nonzero (/ x)) (> (re x) 0))) (ask (= (/ (pow x x)) (pow (/ x) x))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1/2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-252-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle -\left|\sinh x\right| \overset{?}{=} \sinh\left(-\left|x\right|\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0 -> sinh neg abs _0
Lean proof: included
(case sp-acj4-252-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0 -> sinh neg abs _0") (var x complex) (ask (= (- (abs (sinh x))) (sinh (- (abs x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-253-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle -\left|\tanh x\right| \overset{?}{=} \tanh\left(-\left|x\right|\right)\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0 -> tanh neg abs _0
Lean proof: included
(case sp-acj4-253-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0 -> tanh neg abs _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (- (abs (tanh x))) (tanh (- (abs x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-260-complex
Given
\(x \in \mathbb{C},\quad \cos x \neq 0,\quad \frac{1}{\cos x} \neq 0\)
Claim
\(\displaystyle -\log\left(\cos x\right) \overset{?}{=} \log\left(\frac{1}{\cos x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2\)
SimpliPy acj-4 (MIT) rules_real.json[260]: neg log cos _0 -> log inv cos _0
Lean proof: included
(case sp-acj4-260-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[260]: neg log cos _0 -> log inv cos _0") (var x complex) (assume (nonzero (cos x))) (assume (nonzero (/ (cos x)))) (ask (= (- (log (cos x))) (log (/ (cos x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x 2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-261-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \frac{1}{\cosh x} \neq 0\)
Claim
\(\displaystyle -\log\left(\cosh x\right) \overset{?}{=} \log\left(\frac{1}{\cosh x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[261]: neg log cosh _0 -> log inv cosh _0
Lean proof: included
(case sp-acj4-261-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[261]: neg log cosh _0 -> log inv cosh _0") (var x complex) (assume (nonzero (cosh x))) (assume (nonzero (/ (cosh x)))) (ask (= (- (log (cosh x))) (log (/ (cosh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-266-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0\)
Claim
\(\displaystyle -\sin\left(\log x\right) \overset{?}{=} \sin\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[266]: neg sin log _0 -> sin log inv _0
Lean proof: included
(case sp-acj4-266-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[266]: neg sin log _0 -> sin log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (/ x))) (ask (= (- (sin (log x))) (sin (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-267-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \cos\left(\log x\right) \neq 0,\quad \frac{1}{x} \neq 0,\quad \cos\left(\log\left(\frac{1}{x}\right)\right) \neq 0\)
Claim
\(\displaystyle -\tan\left(\log x\right) \overset{?}{=} \tan\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[267]: neg tan log _0 -> tan log inv _0
Lean proof: included
(case sp-acj4-267-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[267]: neg tan log _0 -> tan log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (cos (log x)))) (assume (nonzero (/ x))) (assume (nonzero (cos (log (/ x))))) (ask (= (- (tan (log x))) (tan (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-274-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \left|x\right|^{-2} \overset{?}{=} x^{-2}\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[274] + rules.json[183]: pow abs _0 (-2) -> pow _0 (-2)
Lean proof: included
(case sp-acj4-274-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[274] + rules.json[183]: pow abs _0 (-2) -> pow _0 (-2)") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (pow (abs x) -2) (pow x -2))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-278-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \left|x\right|^{-6} \overset{?}{=} x^{-6}\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[278] + rules.json[187]: pow abs _0 (-6) -> pow _0 (-6)
Lean proof: included
(case sp-acj4-278-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[278] + rules.json[187]: pow abs _0 (-6) -> pow _0 (-6)") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (pow (abs x) -6) (pow x -6))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-283-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right|^{2} \overset{?}{=} x \, x\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2 -> * _0 _0
Lean proof: included
(case sp-acj4-283-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2 -> * _0 _0") (var x complex) (ask (= (pow (abs x) 2) (* x x))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")))
sp-acj4-284-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right|^{3} = \left|x^{3}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[284] + rules.json[193]: pow abs _0 3 -> abs pow _0 3
Lean proof: included
(case sp-acj4-284-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[284] + rules.json[193]: pow abs _0 3 -> abs pow _0 3") (var x complex) (ask (= (pow (abs x) 3) (abs (pow x 3)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-288-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right|^{7} = \left|x^{7}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[288] + rules.json[197]: pow abs _0 7 -> abs pow _0 7
Lean proof: included
(case sp-acj4-288-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[288] + rules.json[197]: pow abs _0 7 -> abs pow _0 7") (var x complex) (ask (= (pow (abs x) 7) (abs (pow x 7)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-30-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle 4 \, \left|x\right| = \left|-4 \, x\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[30] + rules.json[23]: * 4 abs _0 -> abs * (-4) _0
Lean proof: included
(case sp-acj4-30-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[30] + rules.json[23]: * 4 abs _0 -> abs * (-4) _0") (var x complex) (ask (= (* 4 (abs x)) (abs (* -4 x)))) (expect true) (lean-proof "simp [AE.abs_eq]"))
sp-acj4-334-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0 \lor \operatorname{Re}\left(e^{1}\right) > 0\)
Claim
\(\displaystyle \left(e^{x}\right)^{e^{1}} \overset{?}{=} e^{e^{1} \, x}\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[334]: pow exp _0 np.e -> exp * np.e _0
Lean proof: included
(case sp-acj4-334-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[334]: pow exp _0 np.e -> exp * np.e _0") (var x complex) (assume (or (nonzero (exp x)) (> (re (exp 1)) 0))) (ask (= (pow (exp x) (exp 1)) (exp (* (exp 1) x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-335-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0\)
Claim
\(\displaystyle \left(e^{x}\right)^{\pi} \overset{?}{=} e^{\pi \, x}\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[335]: pow exp _0 np.pi -> exp * np.pi _0
Lean proof: included
(case sp-acj4-335-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[335]: pow exp _0 np.pi -> exp * np.pi _0") (var x complex) (assume (or (nonzero (exp x)) (> (re pi) 0))) (ask (= (pow (exp x) pi) (exp (* pi x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-336-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0 \lor \operatorname{Re}\left(x\right) > 0\)
Claim
\(\displaystyle \left(e^{x}\right)^{x} \overset{?}{=} e^{x \, x}\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[336]: pow exp _0 _0 -> exp * _0 _0
Lean proof: included
(case sp-acj4-336-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[336]: pow exp _0 _0 -> exp * _0 _0") (var x complex) (assume (or (nonzero (exp x)) (> (re x) 0))) (ask (= (pow (exp x) x) (exp (* x x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-343-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \frac{1}{\pi}^{x} = \pi^{-x}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[343]: pow inv np.pi _0 -> pow np.pi neg _0
Lean proof: included
(case sp-acj4-343-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[343]: pow inv np.pi _0 -> pow np.pi neg _0") (var x complex) (ask (= (pow (/ pi) x) (pow pi (- x)))) (expect true) (lean-proof "refine ⟨by simp [Real.pi_ne_zero], ?_⟩; simp only [AE.pi_def, AE.inv_def]; exact AE.pow_inv_base Real.pi_pos x"))
sp-acj4-344-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0 \lor \operatorname{Re}\left(x\right) > 0,\quad x \neq 0 \lor \operatorname{Re}\left(-x\right) > 0\)
Claim
\(\displaystyle \frac{1}{x}^{x} \overset{?}{=} x^{-x}\)
Expected
contingent true at \(x = 1\); false at \(x = -\frac{1}{2}\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -\frac{1}{2}\)
SimpliPy acj-4 (MIT) rules_real.json[344]: pow inv _0 _0 -> pow _0 neg _0
Lean proof: included
(case sp-acj4-344-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[344]: pow inv _0 _0 -> pow _0 neg _0") (var x complex) (assume (nonzero x)) (assume (or (nonzero (/ x)) (> (re x) 0))) (assume (or (nonzero x) (> (re (- x)) 0))) (ask (= (pow (/ x) x) (pow x (- x)))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1/2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-368-complex
Given
\(x \in \mathbb{C},\quad \arccos x \in \mathbb{R},\quad \arccos x \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(\arccos x, -5\right) = \left(\arccos x\right)^{-\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5) -> pow acos _0 -0.2
Lean proof: included
(case sp-acj4-368-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5) -> pow acos _0 -0.2") (var x complex) (assume (real (acos x))) (assume (nonzero (acos x))) (ask (= (surd (acos x) -5) (pow (acos x) -1/5))) (expect true) (lean-proof "have hi : (AE.acos x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acos_re_nonneg hi; refine ⟨⟨⟨AE.acos.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨-5, by norm_num⟩ : ∃ n : ℤ, (-5:ℂ) = n), by norm_num, Or.inl hr, Or.inl h1.2⟩⟩, AE.acos.D_true x, (AE.pow.D_iff _ _).2 (Or.inl h1.2)⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]; norm_num"))
sp-acj4-369-complex
Given
\(x \in \mathbb{C},\quad \arccos x \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(\arccos x, 5\right) = \left(\arccos x\right)^{\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5 -> pow acos _0 0.2
Lean proof: included
(case sp-acj4-369-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5 -> pow acos _0 0.2") (var x complex) (assume (real (acos x))) (ask (= (surd (acos x) 5) (pow (acos x) 1/5))) (expect true) (lean-proof "have hi : (AE.acos x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acos_re_nonneg hi; refine ⟨⟨⟨AE.acos.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨5, by norm_num⟩ : ∃ n : ℤ, (5:ℂ) = n), by norm_num, Or.inl hr, Or.inr (by norm_num)⟩⟩, AE.acos.D_true x, (AE.pow.D_iff _ _).2 (Or.inr (by norm_num))⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]"))
sp-acj4-370-complex
Given
\(x \in \mathbb{C},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), -5\right) = \operatorname{arcosh}\left(x\right)^{-\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _0 -0.2
Lean proof: included
(case sp-acj4-370-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _0 -0.2") (var x complex) (assume (real (acosh x))) (assume (nonzero (acosh x))) (ask (= (surd (acosh x) -5) (pow (acosh x) -1/5))) (expect true) (lean-proof "have hi : (AE.acosh x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acosh_re_nonneg hi; refine ⟨⟨⟨AE.acosh.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨-5, by norm_num⟩ : ∃ n : ℤ, (-5:ℂ) = n), by norm_num, Or.inl hr, Or.inl h1.2⟩⟩, AE.acosh.D_true x, (AE.pow.D_iff _ _).2 (Or.inl h1.2)⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]; norm_num"))
sp-acj4-371-complex
Given
\(x \in \mathbb{C},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), 5\right) = \operatorname{arcosh}\left(x\right)^{\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5 -> pow acosh _0 0.2
Lean proof: included
(case sp-acj4-371-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5 -> pow acosh _0 0.2") (var x complex) (assume (real (acosh x))) (ask (= (surd (acosh x) 5) (pow (acosh x) 1/5))) (expect true) (lean-proof "have hi : (AE.acosh x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acosh_re_nonneg hi; refine ⟨⟨⟨AE.acosh.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨5, by norm_num⟩ : ∃ n : ℤ, (5:ℂ) = n), by norm_num, Or.inl hr, Or.inr (by norm_num)⟩⟩, AE.acosh.D_true x, (AE.pow.D_iff _ _).2 (Or.inr (by norm_num))⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]"))
sp-acj4-372-complex
Given
\(x \in \mathbb{C},\quad \cosh x \in \mathbb{R},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(\cosh x, -5\right) \overset{?}{=} \left(\cosh x\right)^{-\frac{1}{5}}\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5) -> pow cosh _0 -0.2
Lean proof: included
(case sp-acj4-372-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5) -> pow cosh _0 -0.2") (var x complex) (assume (real (cosh x))) (assume (nonzero (cosh x))) (ask (= (surd (cosh x) -5) (pow (cosh x) -1/5))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-373-complex
Given
\(x \in \mathbb{C},\quad \cosh x \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(\cosh x, 5\right) \overset{?}{=} \left(\cosh x\right)^{\frac{1}{5}}\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5 -> pow cosh _0 0.2
Lean proof: included
(case sp-acj4-373-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5 -> pow cosh _0 0.2") (var x complex) (assume (real (cosh x))) (ask (= (surd (cosh x) 5) (pow (cosh x) 1/5))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-387-complex
Given
\(x \in \mathbb{C},\quad e^{x} \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(e^{x}, 5\right) \overset{?}{=} e^{\frac{x}{5}}\)
Expected
contingent true at \(x = 0\); false at \(x = \pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = \pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[387]: rootn exp _0 5 -> exp / _0 5
Lean proof: included
(case sp-acj4-387-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[387]: rootn exp _0 5 -> exp / _0 5") (var x complex) (assume (real (exp x))) (ask (= (surd (exp x) 5) (exp (/ x 5)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-389-complex
Given
\(x \in \mathbb{C},\quad e^{x} \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{surd}\left(e^{x}, 7\right) \overset{?}{=} e^{\frac{x}{7}}\)
Expected
contingent true at \(x = 0\); false at \(x = \pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = \pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[389]: rootn exp _0 7 -> exp / _0 7
Lean proof: included
(case sp-acj4-389-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[389]: rootn exp _0 7 -> exp / _0 7") (var x complex) (assume (real (exp x))) (ask (= (surd (exp x) 7) (exp (/ x 7)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-396-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \sin\left(\left|\sin x\right|\right) \overset{?}{=} \left|\sin\left(\sin x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0 -> abs sin sin _0
Lean proof: included
(case sp-acj4-396-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0 -> abs sin sin _0") (var x complex) (ask (= (sin (abs (sin x))) (abs (sin (sin x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-40-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \left|x\right| \, \pi = \left|\pi \, x\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[40] + rules.json[33]: * abs _0 np.pi -> abs * np.pi _0
Lean proof: included
(case sp-acj4-40-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[40] + rules.json[33]: * abs _0 np.pi -> abs * np.pi _0") (var x complex) (ask (= (* (abs x) pi) (abs (* pi x)))) (expect true) (lean-proof "simp [AE.abs_eq]; ring"))
sp-acj4-403-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0\)
Claim
\(\displaystyle \sin\left(-\log x\right) \overset{?}{=} \sin\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[403]: sin neg log _0 -> sin log inv _0
Lean proof: included
(case sp-acj4-403-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[403]: sin neg log _0 -> sin log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (/ x))) (ask (= (sin (- (log x))) (sin (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-404-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \sin\left(\tanh\left(\left|x\right|\right)\right) \overset{?}{=} \left|\sin\left(\tanh x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0 -> abs sin tanh _0
Lean proof: included
(case sp-acj4-404-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0 -> abs sin tanh _0") (var x complex) (assume (nonzero (cosh x))) (ask (= (sin (tanh (abs x))) (abs (sin (tanh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-414-complex
Given
\(x \in \mathbb{C},\quad \cos\left(\left|\sin x\right|\right) \neq 0,\quad \cos\left(\sin x\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\left|\sin x\right|\right) \overset{?}{=} \left|\tan\left(\sin x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0 -> abs tan sin _0
Lean proof: included
(case sp-acj4-414-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0 -> abs tan sin _0") (var x complex) (assume (nonzero (cos (abs (sin x))))) (assume (nonzero (cos (sin x)))) (ask (= (tan (abs (sin x))) (abs (tan (sin x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-415-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \cos\left(\left|\tanh x\right|\right) \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\left|\tanh x\right|\right) \overset{?}{=} \left|\tan\left(\tanh x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0 -> abs tan tanh _0
Lean proof: included
(case sp-acj4-415-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0 -> abs tan tanh _0") (var x complex) (assume (nonzero (cosh x))) (assume (nonzero (cos (abs (tanh x))))) (assume (nonzero (cos (tanh x)))) (ask (= (tan (abs (tanh x))) (abs (tan (tanh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-420-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \cos\left(-\log x\right) \neq 0,\quad \frac{1}{x} \neq 0,\quad \cos\left(\log\left(\frac{1}{x}\right)\right) \neq 0\)
Claim
\(\displaystyle \tan\left(-\log x\right) \overset{?}{=} \tan\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[420]: tan neg log _0 -> tan log inv _0
Lean proof: included
(case sp-acj4-420-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[420]: tan neg log _0 -> tan log inv _0") (var x complex) (assume (nonzero x)) (assume (nonzero (cos (- (log x))))) (assume (nonzero (/ x))) (assume (nonzero (cos (log (/ x))))) (ask (= (tan (- (log x))) (tan (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-421-complex
Given
\(x \in \mathbb{C},\quad \cos\left(\tanh\left(\left|x\right|\right)\right) \neq 0,\quad \cosh x \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\tanh\left(\left|x\right|\right)\right) \overset{?}{=} \left|\tan\left(\tanh x\right)\right|\)
Expected
contingent true at \(x = 0\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0 -> abs tan tanh _0
Lean proof: included
(case sp-acj4-421-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0 -> abs tan tanh _0") (var x complex) (assume (nonzero (cos (tanh (abs x))))) (assume (nonzero (cosh x))) (assume (nonzero (cos (tanh x)))) (ask (= (tan (tanh (abs x))) (abs (tan (tanh x))))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-422-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \tanh\left(\left|\operatorname{arsinh}\left(x\right)\right|\right) \overset{?}{=} \cos\left(\arctan\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[422]: tanh abs asinh _0 -> cos atan inv _0
Lean proof: included
(case sp-acj4-422-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[422]: tanh abs asinh _0 -> cos atan inv _0") (var x complex) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (tanh (abs (asinh x))) (cos (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-423-complex
Given
\(x \in \mathbb{C},\quad \cosh\left(\operatorname{arcosh}\left(\left|x\right|\right)\right) \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \tanh\left(\operatorname{arcosh}\left(\left|x\right|\right)\right) \overset{?}{=} \cos\left(\arcsin\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = i\)
SimpliPy acj-4 (MIT) rules_real.json[423]: tanh acosh abs _0 -> cos asin inv _0
Lean proof: included
(case sp-acj4-423-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[423]: tanh acosh abs _0 -> cos asin inv _0") (var x complex) (assume (nonzero (cosh (acosh (abs x))))) (assume (nonzero x)) (ask (= (tanh (acosh (abs x))) (cos (asin (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-424-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq i,\quad \frac{1}{x} \neq -i\)
Claim
\(\displaystyle \tanh\left(\operatorname{arsinh}\left(\left|x\right|\right)\right) \overset{?}{=} \cos\left(\arctan\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[424]: tanh asinh abs _0 -> cos atan inv _0
Lean proof: included
(case sp-acj4-424-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[424]: tanh asinh abs _0 -> cos atan inv _0") (var x complex) (assume (nonzero x)) (assume (!= (/ x) i)) (assume (!= (/ x) (- i))) (ask (= (tanh (asinh (abs x))) (cos (atan (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-430-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(x\right) > 0,\quad x^{x} \neq 0\)
Claim
\(\displaystyle \log x \, x \overset{?}{=} \log\left(x^{x}\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[430]: * log ?0 ?0 -> log pow ?0 ?0
Lean proof: included
(case sp-acj4-430-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[430]: * log ?0 ?0 -> log pow ?0 ?0") (var x complex) (assume (nonzero x)) (assume (or (nonzero x) (> (re x) 0))) (assume (nonzero (pow x x))) (ask (= (* (log x) x) (log (pow x x)))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-433-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad -\log x \neq i,\quad -\log x \neq -i,\quad \frac{1}{x} \neq 0,\quad \log\left(\frac{1}{x}\right) \neq i,\quad \log\left(\frac{1}{x}\right) \neq -i\)
Claim
\(\displaystyle \arctan\left(-\log x\right) \overset{?}{=} \arctan\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[433]: atan neg log !0 -> atan log inv !0
Lean proof: included
(case sp-acj4-433-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[433]: atan neg log !0 -> atan log inv !0") (var x complex) (assume (nonzero x)) (assume (!= (- (log x)) i)) (assume (!= (- (log x)) (- i))) (assume (nonzero (/ x))) (assume (!= (log (/ x)) i)) (assume (!= (log (/ x)) (- i))) (ask (= (atan (- (log x))) (atan (log (/ x))))) (expect contingent) (witness true (x 1) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x -2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234]; exact AE.w1_c433")))
sp-acj4-442-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad \log x \neq 0,\quad \frac{1}{\log x} \neq 0\)
Claim
\(\displaystyle -\log\left(\log x\right) \overset{?}{=} \log\left(\frac{1}{\log x}\right)\)
Expected
contingent true at \(x = -1\); false at \(x = \frac{1}{2}\)
equal at some admissible points, different at others
witness (true): \(x = -1\)
witness (false): \(x = \frac{1}{2}\)
SimpliPy acj-4 (MIT) rules_real.json[442]: neg log log ?0 -> log inv log ?0
Lean proof: included
(case sp-acj4-442-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[442]: neg log log ?0 -> log inv log ?0") (var x complex) (assume (nonzero x)) (assume (nonzero (log x))) (assume (nonzero (/ (log x)))) (ask (= (- (log (log x))) (log (/ (log x))))) (expect contingent) (witness true (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x 1/2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-446-complex
Given
\(x \in \mathbb{C},\quad x \neq i,\quad x \neq -i,\quad \cos\left(\left|\arctan x\right|\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\left|\arctan x\right|\right) \overset{?}{=} \left|x\right|\)
Expected
contingent true at \(x = 0\); false at \(x = \frac{1}{2} \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = \frac{1}{2} \, i\)
SimpliPy acj-4 (MIT) rules_real.json[446]: tan abs atan !0 -> abs !0
Lean proof: included
(case sp-acj4-446-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[446]: tan abs atan !0 -> abs !0") (var x complex) (assume (!= x i)) (assume (!= x (- i))) (assume (nonzero (cos (abs (atan x))))) (ask (= (tan (abs (atan x))) (abs x))) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I']; done) | (norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', Complex.ext_iff]; done)")) (witness false (x (* 1/2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-450-complex
Given
\(x \in \mathbb{C},\quad x \neq 0\)
Claim
\(\displaystyle \frac{x}{x} = 1\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[450]: / $0 $0 -> 1
Lean proof: included
(case sp-acj4-450-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[450]: / $0 $0 -> 1") (var x complex) (assume (nonzero x)) (ask (= (/ x x) 1)) (expect true) (lean-proof "simp_all"))
sp-acj4-451-complex
Given
\(x \in \mathbb{C},\quad x \neq 0\)
Claim
\(\displaystyle \frac{0}{x} = 0\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[451]: / 0 $0 -> 0
Lean proof: included
(case sp-acj4-451-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[451]: / 0 $0 -> 0") (var x complex) (assume (nonzero x)) (ask (= (/ 0 x) 0)) (expect true) (lean-proof "simp_all"))
sp-acj4-50-complex
Given
\(x \in \mathbb{C},\quad 6^{x} \neq 0\)
Claim
\(\displaystyle \log 6 \, x \overset{?}{=} \log\left(6^{x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[50]: * log 6 _0 -> log pow 6 _0
Lean proof: included
(case sp-acj4-50-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[50]: * log 6 _0 -> log pow 6 _0") (var x complex) (assume (nonzero (pow 6 x))) (ask (= (* (log 6) x) (log (pow 6 x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-51-complex
Given
\(x \in \mathbb{C},\quad 7^{x} \neq 0\)
Claim
\(\displaystyle \log 7 \, x \overset{?}{=} \log\left(7^{x}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[51]: * log 7 _0 -> log pow 7 _0
Lean proof: included
(case sp-acj4-51-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[51]: * log 7 _0 -> log pow 7 _0") (var x complex) (assume (nonzero (pow 7 x))) (ask (= (* (log 7) x) (log (pow 7 x)))) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-55-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(e^{1}\right) > 0,\quad x^{e^{1}} \neq 0\)
Claim
\(\displaystyle \log x \, e^{1} \overset{?}{=} \log\left(x^{e^{1}}\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[55]: * log _0 np.e -> log pow _0 np.e
Lean proof: included
(case sp-acj4-55-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[55]: * log _0 np.e -> log pow _0 np.e") (var x complex) (assume (nonzero x)) (assume (or (nonzero x) (> (re (exp 1)) 0))) (assume (nonzero (pow x (exp 1)))) (ask (= (* (log x) (exp 1)) (log (pow x (exp 1))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-56-complex
Given
\(x \in \mathbb{C},\quad x \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0,\quad x^{\pi} \neq 0\)
Claim
\(\displaystyle \log x \, \pi \overset{?}{=} \log\left(x^{\pi}\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -1\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -1\)
SimpliPy acj-4 (MIT) rules_real.json[56]: * log _0 np.pi -> log pow _0 np.pi
Lean proof: included
(case sp-acj4-56-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[56]: * log _0 np.pi -> log pow _0 np.pi") (var x complex) (assume (nonzero x)) (assume (or (nonzero x) (> (re pi) 0))) (assume (nonzero (pow x pi))) (ask (= (* (log x) pi) (log (pow x pi)))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x -1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-6-complex
Given
\(x \in \mathbb{C},\quad \cosh x \neq 0,\quad \tanh x \neq 1,\quad \tanh x \neq -1\)
Claim
\(\displaystyle \operatorname{artanh}\left(\tanh x\right) \overset{?}{=} x\)
Expected
contingent true at \(x = 0\); false at \(x = 2 \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = 2 \, i\)
SimpliPy acj-4 (MIT) rules_real.json[6]: atanh tanh !0 -> !0
Lean proof: included
(case sp-acj4-6-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[6]: atanh tanh !0 -> !0") (var x complex) (assume (nonzero (cosh x))) (assume (!= (tanh x) 1)) (assume (!= (tanh x) -1)) (ask (= (atanh (tanh x)) x)) (expect contingent) (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")) (witness false (x (* 2 i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-7-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{1}{\left|x\right|} = \left|\frac{1}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[7] + rules.json[4]: inv abs _0 -> abs inv _0
Lean proof: included
(case sp-acj4-7-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[7] + rules.json[4]: inv abs _0 -> abs inv _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ (abs x)) (abs (/ x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-9-complex
Given
\(x \in \mathbb{C},\quad e^{x} \neq 0\)
Claim
\(\displaystyle \log\left(e^{x}\right) \overset{?}{=} x\)
Expected
contingent true at \(x = 0\); false at \(x = -\pi \, i\)
equal at some admissible points, different at others
witness (true): \(x = 0\)
witness (false): \(x = -\pi \, i\)
SimpliPy acj-4 (MIT) rules_real.json[9]: log exp _0 -> _0
Lean proof: included
(case sp-acj4-9-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[9]: log exp _0 -> _0") (var x complex) (assume (nonzero (exp x))) (ask (= (log (exp x)) x)) (expect contingent) (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)")) (witness false (x (* -1 pi i)) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))
sp-acj4-90-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{3}{\left|x\right|} = \left|\frac{-3}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0 -> abs / (-3) _0
Lean proof: included
(case sp-acj4-90-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0 -> abs / (-3) _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ 3 (abs x)) (abs (/ -3 x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-94-complex
Given
\(x \in \mathbb{C},\quad \left|x\right| \neq 0,\quad x \neq 0\)
Claim
\(\displaystyle \frac{7}{\left|x\right|} = \left|\frac{-7}{x}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[94] + rules.json[73]: / 7 abs _0 -> abs / (-7) _0
Lean proof: included
(case sp-acj4-94-complex (category simplipy-complex-branch) (note "SimpliPy acj-4 (MIT) rules_real.json[94] + rules.json[73]: / 7 abs _0 -> abs / (-7) _0") (var x complex) (assume (nonzero (abs x))) (assume (nonzero x)) (ask (= (/ 7 (abs x)) (abs (/ -7 x)))) (expect true) (lean-proof "simp [AE.abs_eq] at *; exact h1"))
sp-acj4-99-complex
Given
\(x \in \mathbb{C}\)
Claim
\(\displaystyle \frac{\left|x\right|}{3} = \left|\frac{x}{-3}\right|\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3 -> abs / _0 (-3)
Lean proof: included