Proof: Substitute Add 4

Let's prove the following theorem:

if the following are true:
  • value of cell 3 at time 5 = (value of cell 1 at time 4) + (value of cell 2 at time 4)
  • value of cell 1 at time 4 = 0
  • value of cell 2 at time 4 = 1

then value of cell 3 at time 5 = 1

Proof:

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Given
1 value of cell 3 at time 5 = (value of cell 1 at time 4) + (value of cell 2 at time 4)
2 value of cell 1 at time 4 = 0
3 value of cell 2 at time 4 = 1
Proof Table
# Claim Reason
1 (value of cell 1 at time 4) + (value of cell 2 at time 4) = 0 + (value of cell 2 at time 4) if value of cell 1 at time 4 = 0, then (value of cell 1 at time 4) + (value of cell 2 at time 4) = 0 + (value of cell 2 at time 4)
2 value of cell 3 at time 5 = 0 + (value of cell 2 at time 4) if value of cell 3 at time 5 = (value of cell 1 at time 4) + (value of cell 2 at time 4) and (value of cell 1 at time 4) + (value of cell 2 at time 4) = 0 + (value of cell 2 at time 4), then value of cell 3 at time 5 = 0 + (value of cell 2 at time 4)
3 0 + (value of cell 2 at time 4) = 0 + 1 if value of cell 2 at time 4 = 1, then 0 + (value of cell 2 at time 4) = 0 + 1
4 value of cell 3 at time 5 = 0 + 1 if value of cell 3 at time 5 = 0 + (value of cell 2 at time 4) and 0 + (value of cell 2 at time 4) = 0 + 1, then value of cell 3 at time 5 = 0 + 1
5 0 + 1 = 1 0 + 1 = 1
6 value of cell 3 at time 5 = 1 if value of cell 3 at time 5 = 0 + 1 and 0 + 1 = 1, then value of cell 3 at time 5 = 1

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