Proof: Append to List Example

Let's prove the following theorem:

In step 2, we state that the reverse of

[1, [2, [3, []]]]

is

[3,[2,[1,[]]]]

Then we prepend 4 to this list in step 3, which leads to:

[4,[3,[2,[1,[]]]]]

Then step 5 shows that the reverse of the above list is:

[1,[2,[3,[4,[]]]]]

Proof:

Proof Table

#	Claim	Reason
1	result of appending 4 to [ 1, [ 2, [ 3, [ ] ] ] ] = reverse of [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ]	result of appending 4 to [ 1, [ 2, [ 3, [ ] ] ] ] = reverse of [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] (Appending to a List Property)
2	reverse of [ 1, [ 2, [ 3, [ ] ] ] ] = [ 3, [ 2, [ 1, [ ] ] ] ]	reverse of [ 1, [ 2, [ 3, [ ] ] ] ] = [ 3, [ 2, [ 1, [ ] ] ] ] (reverse_list_example_2)
3	[ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] = [ 4, [ 3, [ 2, [ 1, [ ] ] ] ] ]	if reverse of [ 1, [ 2, [ 3, [ ] ] ] ] = [ 3, [ 2, [ 1, [ ] ] ] ], then [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] = [ 4, [ 3, [ 2, [ 1, [ ] ] ] ] ] (Substitution)
4	reverse of [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] = reverse of [ 4, [ 3, [ 2, [ 1, [ ] ] ] ] ]	if [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] = [ 4, [ 3, [ 2, [ 1, [ ] ] ] ] ], then reverse of [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] = reverse of [ 4, [ 3, [ 2, [ 1, [ ] ] ] ] ] (Substitution)
5	reverse of [ 4, [ 3, [ 2, [ 1, [ ] ] ] ] ] = [ 1, [ 2, [ 3, [ 4, [ ] ] ] ] ]	reverse of [ 4, [ 3, [ 2, [ 1, [ ] ] ] ] ] = [ 1, [ 2, [ 3, [ 4, [ ] ] ] ] ] (reverse_list_example_3)
6	reverse of [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] = [ 1, [ 2, [ 3, [ 4, [ ] ] ] ] ]	if reverse of [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] = reverse of [ 4, [ 3, [ 2, [ 1, [ ] ] ] ] ] and reverse of [ 4, [ 3, [ 2, [ 1, [ ] ] ] ] ] = [ 1, [ 2, [ 3, [ 4, [ ] ] ] ] ], then reverse of [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] = [ 1, [ 2, [ 3, [ 4, [ ] ] ] ] ] (Transitive Property of Equality)
7	result of appending 4 to [ 1, [ 2, [ 3, [ ] ] ] ] = [ 1, [ 2, [ 3, [ 4, [ ] ] ] ] ]	if result of appending 4 to [ 1, [ 2, [ 3, [ ] ] ] ] = reverse of [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] and reverse of [ 4, reverse of [ 1, [ 2, [ 3, [ ] ] ] ] ] = [ 1, [ 2, [ 3, [ 4, [ ] ] ] ] ], then result of appending 4 to [ 1, [ 2, [ 3, [ ] ] ] ] = [ 1, [ 2, [ 3, [ 4, [ ] ] ] ] ] (Transitive Property of Equality)

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