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
[ 1, [ 0, [ 1, [ ] ] ] ] is less than [ 0, [ 1, [ 1, [ ] ] ] ]
minimum value of stack [ [ 0, [ ] ], [ [ 1, [ ] ], [ ] ] ] = [ 0, [ ] ]
minimum value of stack [ [ 1, [ ] ], [ [ 0, [ ] ], [ ] ] ] = [ 0, [ ] ]
minimum value of stack [ [ 1, [ 1, [ ] ] ], [ [ 0, [ 1, [ ] ] ], [ [ 0, [ 0, [ ] ] ], [ ] ] ] ] = [ 0, [ 0, [ ] ] ]
[0,[]] = [0,[]]
bit [ 0, [ 1, [ ] ] ] minus bit 1 = [ 1, [ 0, [ ] ] ]
result of dumping [ [ 1, [ ] ], [ [ 0, [ ] ], [ ] ] ] to [ [ 0, [ ] ], [ ] ] = [ [ 0, [ ] ], [ [ 1, [ ] ], [ [ 0, [ ] ], [ ] ] ] ]
remaining elements after [ [ 0, [ ] ], [ ] ] is popped at index [ 0, [ ] ] = [ ]
remaining elements after [ [ x, [ ] ], [ ] ] is popped at index [ 0, [ ] ] = [ ]
remaining elements after [ [ 0, [ ] ], [ [ 1, [ ] ], [ ] ] ] is popped at index [ 1, [ ] ] and visited stack is [ ] = [ [ 0, [ ] ], [ ] ]
remaining elements after [ [ 0, [ ] ], [ [ 1, [ ] ], [ ] ] ] is popped at index [ 1, [ ] ] = [ [ 0, [ ] ], [ ] ]
remaining elements after [ x, [ y, [ ] ] ] is popped at index [ 1, [ ] ] and visited stack is [ ] = [ x, [ ] ]
remaining elements after [ x, [ y, [ ] ] ] is popped at index [ 1, [ ] ] = [ x, [ ] ]
index of value [ 0, [ ] ] in [ [ 0, [ ] ], [ ] ] = [ 0, [ ] ]
index of value [ 0, [ ] ] in [ [ 1, [ ] ], [ [ 0, [ ] ], [ ] ] ] = [ 1, [ ] ]
index of the mininum value in stack [ x, [ ] ] = [ 0, [ ] ]
index of the mininum value in stack [ [ 1, [ ] ], [ [ 0, [ ] ], [ ] ] ] = [ 1, [ ] ]
result of sorting [ [ 0, [ ] ], [ ] ] = [ [ 0, [ ] ], [ ] ]
result of sorting [ [ 1, [ ] ], [ [ 0, [ ] ], [ ] ] ] = [ [ 0, [ ] ], [ [ 1, [ ] ], [ ] ] ]
if the following are true:
- a < b
- a = x
- b = y
then x < y
if the following are true:
- a < b
- x = a
- y = b
then x < y
reverse of [ x, [ ] ] = [ x, [ ] ]
reverse of [ x, [ y, [ ] ] ] = [ y, [ x, [ ] ] ]
reverse of [ 1, [ 2, [ 3, [ 4, [ ] ] ] ] ] = [ 4, [ 3, [ 2, [ 1, [ ] ] ] ] ]
minimum value of stack [ 3, [ 2, [ 1, [ ] ] ] ] = 1
index of value x in [ x, [ ] ] = 0
index of value 1 in [ 3, [ 2, [ 1, [ ] ] ] ] = 2
index of the mininum value in stack [ 3, [ 2, [ 1, [ ] ] ] ] = 2
remaining elements after [ x, [ ] ] is popped at index 0 = [ ]
remaining elements after [ 3, [ 2, [ 1, [ ] ] ] ] is popped at index 2 and visited stack is [ ] = [ 3, [ 2, [ ] ] ]
remaining elements after [ 3, [ 2, [ 1, [ ] ] ] ] is popped at index 2 = [ 3, [ 2, [ ] ] ]
maximum value in stack [ 2, [ 1, [ ] ] ] = 2
maximum value in stack [ 3, [ 2, [ 1, [ ] ] ] ] = 3
index of value 3 in [ 3, [ 2, [ 1, [ ] ] ] ] = 0
index of the maximum value in stack [ x, [ ] ] = 0
index of the maximum value in stack [ 3, [ 2, [ 1, [ ] ] ] ] = 0
result of dumping [ 2, [ 1, [ ] ] ] to [ ] = [ 1, [ 2, [ ] ] ]
remaining elements after [ 3, [ 2, [ 1, [ ] ] ] ] is popped at index 0 and visited stack is [ ] = [ 2, [ 1, [ ] ] ]
remaining elements after [ 3, [ 2, [ 1, [ ] ] ] ] is popped at index 0 = [ 2, [ 1, [ ] ] ]
index of value 2 in [ 2, [ 1, [ ] ] ] = 0
index of the maximum value in stack [ 2, [ 1, [ ] ] ] = 0
remaining elements after [ 2, [ 1, [ ] ] ] is popped at index 0 and visited stack is [ ] = [ 1, [ ] ]
remaining elements after [ 2, [ 1, [ ] ] ] is popped at index 0 = [ 1, [ ] ]
result of sorting [ 3, [ 2, [ 1, [ ] ] ] ] = [ 1, [ 2, [ 3, [ ] ] ] ]
if the following are true:
- the PC at time 0 = 0
- instruction #0 is
addi dst=1 src=0 imm=0 - instruction #1 is
addi dst=2 src=0 imm=1 - instruction #2 is
addi dst=4 src=0 imm=0 - instruction #3 is
addi dst=5 src=0 imm=4 - instruction #4 is
add dst=3 src1=1 src2=2 - instruction #5 is
addi dst=1 src=2 imm=0 - instruction #6 is
addi dst=2 src=3 imm=0 - instruction #7 is
addi dst=4 src=4 imm=1 - instruction #8 is
beq left=4 right=5 imm=1 - instruction #9 is
jump imm=4
then value of cell 3 at time 25 = 5
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