conventions
Case set df65d778fc5c
demo-acot-01
Given
\(x \in \mathbb{R},\quad x > 0\)
Claim
\(\displaystyle \operatorname{arccot}\left(x\right) = \arctan\left(\frac{1}{x}\right)\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
The acot conventions agree on the positive reals.
(case demo-acot-01 (category conventions) (note "The acot conventions agree on the positive reals.") (var x real) (assume (positive x)) (ask (= (acot x) (atan (/ 1 x)))) (expect true))
demo-heaviside-01
Given
\(x \in \mathbb{R},\quad x \neq 0\)
Claim
\(\displaystyle H\left(x\right) + H\left(-x\right) = 1\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
Holds under every convention for heaviside(0), because x = 0 is excluded.
(case demo-heaviside-01 (category conventions) (note "Holds under every convention for heaviside(0), because x = 0 is excluded.") (var x real) (assume (nonzero x)) (ask (= (+ (heaviside x) (heaviside (- x))) 1)) (expect true))
Case set 03f5149718d0
demo-acot-01
Given
\(x \in \mathbb{R},\quad x > 0\)
Claim
\(\displaystyle \operatorname{arccot}\left(x\right) = \arctan\left(\frac{1}{x}\right)\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
The acot conventions agree on the positive reals.
(case demo-acot-01 (category conventions) (note "The acot conventions agree on the positive reals.") (var x real) (assume (positive x)) (ask (= (acot x) (atan (/ 1 x)))) (expect true))
demo-heaviside-01
Given
\(x \in \mathbb{R},\quad x \neq 0\)
Claim
\(\displaystyle H\left(x\right) + H\left(-x\right) = 1\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
Holds under every convention for heaviside(0), because x = 0 is excluded.
(case demo-heaviside-01 (category conventions) (note "Holds under every convention for heaviside(0), because x = 0 is excluded.") (var x real) (assume (nonzero x)) (ask (= (+ (heaviside x) (heaviside (- x))) 1)) (expect true))
Case set d7af4def9aaa
demo-acot-01
Given
\(x \in \mathbb{R},\quad x > 0\)
Claim
\(\displaystyle \operatorname{arccot}\left(x\right) = \arctan\left(\frac{1}{x}\right)\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
The acot conventions agree on the positive reals.
(case demo-acot-01 (category conventions) (note "The acot conventions agree on the positive reals.") (var x real) (assume (positive x)) (ask (= (acot x) (atan (/ 1 x)))) (expect true))
demo-heaviside-01
Given
\(x \in \mathbb{R},\quad x \neq 0\)
Claim
\(\displaystyle H\left(x\right) + H\left(-x\right) = 1\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
Holds under every convention for heaviside(0), because x = 0 is excluded.