Proof: Beq Branch

Let's prove the following theorem:

if the following are true:
  • instruction #5 is beq left=3 right=4 imm=1
  • the PC at time 19 = 5
  • value of cell 3 at time 19 = 3
  • value of cell 4 at time 19 = 3

then the PC at time 20 = 7

Proof:

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Given
1 instruction #5 is beq left=3 right=4 imm=1
2 the PC at time 19 = 5
3 value of cell 3 at time 19 = 3
4 value of cell 4 at time 19 = 3
Proof Table
# Claim Reason
1 value of cell 3 at time 19 = value of cell 4 at time 19 if value of cell 3 at time 19 = 3 and value of cell 4 at time 19 = 3, then value of cell 3 at time 19 = value of cell 4 at time 19
2 the PC at time (19 + 1) = (5 + 1) + 1 if instruction #5 is beq left=3 right=4 imm=1 and the PC at time 19 = 5 and value of cell 3 at time 19 = value of cell 4 at time 19, then the PC at time (19 + 1) = (5 + 1) + 1
3 (5 + 1) + 1 = 7 (5 + 1) + 1 = 7
4 the PC at time (19 + 1) = 7 if the PC at time (19 + 1) = (5 + 1) + 1 and (5 + 1) + 1 = 7, then the PC at time (19 + 1) = 7
5 19 + 1 = 20 19 + 1 = 20
6 the PC at time (19 + 1) = the PC at time 20 if 19 + 1 = 20, then the PC at time (19 + 1) = the PC at time 20
7 the PC at time 20 = 7 if the PC at time (19 + 1) = the PC at time 20 and the PC at time (19 + 1) = 7, then the PC at time 20 = 7

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