Proofs
A proof is a series of claims that lead to a conclusion. Some proofs are conditional, which means that the claims can only be made under certain conditions. Click on a statement to see the proof
(a ⋅ (s / 2)) / s = a / 2
if b > 0, then (c - c) / ((b + a) - a) = 0
if the following are true:
- a = b + c
- x = y + z
- a = x
- b = z
then c = y
if the following are true:
- a = b + c
- b = x
- c = y
then a = x + y
if a = b + c, then a = c + b
if a + b = c, then b + a = c
if the following are true:
- a = b + c
- c = d
then a = b + d
if the following are true:
- a = b + c
- d = c
then a = b + d
if a = b, then (a ⋅ c) + d = (b ⋅ c) + d
if a = b, then a + (b ⋅ (-1)) = 0
if a = b, then c + a = c + b
if a = b, then c + b = c + a
if a = b, then b + c = a + c
if the following are true:
- a = b
- c = d
then a + c = b + d
if x = y, then (a + x) + c = (a + y) + c
if the following are true:
- x = y
- (a + x) + c = f
then (a + y) + c = f
if the following are true:
- a = b
- c = d
then a - c = b - d
if a = b, then b - c = a - c
if a = b, then a - c = b - c
if the following are true:
- a = b
- f = b - c
then f = a - c
if the following are true:
- a = b
- c = d
- f = a - c
then f = b - d
if a = b, then (a ⋅ c) - d = (b ⋅ c) - d
if a = b, then c ⋅ b = c ⋅ a
if a = b, then c ⋅ a = c ⋅ b
if the following are true:
- a = b ⋅ c
- b = d
then a = d ⋅ c
if the following are true:
- a = b ⋅ c
- c = d
then a = b ⋅ d
if the following are true:
- a = x
- b = x ⋅ y
then b = a ⋅ y
if the following are true:
- x = a
- x ⋅ y = b
then a ⋅ y = b
if the following are true:
- a ⋅ b = c
- b = d
then a ⋅ d = c
if a = b, then a / c = b / c
if the following are true:
- a = x
- c = y
then (a + c) / m = (x + y) / m
if the following are true:
- a = x
- c = y
- s = (a + c) / m
then s = (x + y) / m
if a = b + c, then a + (c ⋅ (-1)) = b
if a = b + c, then a + (b ⋅ (-1)) = c
if the following are true:
- a = b + c
- a = d
then d = b + c
if the following are true:
- a = b + c
- b = d
then a = d + c
if the following are true:
- a = b + c
- d = b
then a = d + c
if a = 360 + (180 ⋅ (-1)), then a = 180
if y = 180 + (90 ⋅ (-1)), then y = 90
if 180 + (90 ⋅ (-1)) = y, then 90 = y
if a + 90 = 180, then a = 90
if 90 + a = 180, then a = 90
if 180 = 90 + a, then a = 90
if a = 180 ⋅ (1 / 2), then a = 90
if the following are true:
- a = 90
- b = 90
then a + b = 180
if the following are true:
- a = ((b + c) + d) + e
- d + e = f
then a = (b + c) + f
if the following are true:
- a + (b ⋅ (-1)) = c
- b = d
then a + (d ⋅ (-1)) = c
if a + b = c, then b = c + (a ⋅ (-1))
if the following are true:
- a + b = c
- a = d
then d + b = c
if the following are true:
- a + b = c
- d = a
then d + b = c
if the following are true:
- a + b = c
- b = d
then a + d = c
if the following are true:
- a + b = c
- d = b
then a + d = c
if a + b = c, then a = c + (b ⋅ (-1))
if a + b = c, then b = c + (a ⋅ (-1))
if a + b = c, then a = c - b
if a + b = c, then b = c - a
if the following are true:
- (a + b) + c = d
- a = e
then (e + b) + c = d
if (a + b) + c = d, then b + c = d + (a ⋅ (-1))
if a + b = c, then ((x + a) + b) + y = (x + c) + y
if (a + b) + c = d, then a + c = d - b
if the following are true:
- a / b = c / d
- d = e
then a / b = c / e
if the following are true:
- a = b / c
- b = d
then a = d / c
if the following are true:
- a = x
- b = y
then a / b = x / y
if the following are true:
- a / b = c
- d = a
then c = d / b
if the following are true:
- c = a / b
- d = a
then c = d / b
if the following are true:
- a / b = c
- d = a
then d / b = c
if the following are true:
- a = w
- b = x
- c = y
- d = z
then (a - b) / (c - d) = (w - x) / (y - z)
if the following are true:
- a = b + c
- b > 0
then a > c
if the following are true:
- f = (a - b) / (c - d)
- a = w
- b = x
- c = y
- d = z
then f = (w - x) / (y - z)
sum of (list 0 and (empty list)) (list 0 and (empty list)) and carry bit 0 = list 0 and (empty list)
sum of unsigned integers (list 1 and (empty list)) and (list 1 and (empty list)) = list 0 and (list 1 and (empty list))
sum of unsigned integers (list 1 and (list 1 and (empty list))) and (list 1 and (list 1 and (empty list))) = list 0 and (list 1 and (list 1 and (empty list)))
sum of unsigned integers (list 1 and (list 1 and (empty list))) and (list 0 and (empty list)) = list 1 and (list 1 and (empty list))
sum of unsigned integers (list x and xs) and (list 0 and (empty list)) = list x and xs
(list 1 and (list 1 and (empty list))) multiplied by (list 0 and (list 1 and (empty list))) = list 0 and (list 1 and (list 1 and (empty list)))
(list x and xs) multiplied by (list 0 and (list 1 and (empty list))) = sum of unsigned integers (list x and xs) and (list x and xs)
a + (b + c) = (a + b) + c
0 + a = a
(a ⋅ (-1)) + a = 0
((-1) ⋅ a) + a = 0
a + (b ⋅ (-1)) = a - b
a - a = 0
a - 0 = a
(a + b) - c = a + (b - c)
(b + a) - a = b
a = a ⋅ 1
1 ⋅ a = a
(b + c) ⋅ a = (a ⋅ b) + (a ⋅ c)
(a + b) ⋅ c = (a ⋅ c) + (b ⋅ c)
(a ⋅ b) + (a ⋅ c) = a ⋅ (b + c)
(a ⋅ c) + (b ⋅ c) = (a + b) ⋅ c
a ⋅ (1 / b) = a / b
a ⋅ (b / c) = (a ⋅ b) / c
(a ⋅ b) / c = a ⋅ (b / c)
a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c
(a ⋅ b) ⋅ c = (a ⋅ c) ⋅ b
(a ⋅ b) ⋅ c = (b ⋅ c) ⋅ a
(a - b) ⋅ c = (a ⋅ c) - (b ⋅ c)
((a + b) + c) + d = a + ((b + c) + d)
(a + b) + c = (a + c) + b