Proof: BEQ Instruction Example

Let's prove the following theorem:

if the following are true:
  • instruction #0 is beq left=1 right=2 imm=2
  • the PC at time 0 = 0
  • value of cell 1 at time 0 = 2
  • value of cell 2 at time 0 = 2

then the PC at time 1 = 3

The following example shows that when the value stored in the left cell (cell #1) is equal to the value in the right cell (cell #2), the PC is incremented by imm (2) + 1.

Instructions
Memory Cells
Program Counter Time
0 0
LW Computer Simulator

Proof:

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Given
1 instruction #0 is beq left=1 right=2 imm=2
2 the PC at time 0 = 0
3 value of cell 1 at time 0 = 2
4 value of cell 2 at time 0 = 2
Proof Table
# Claim Reason
1 value of cell 1 at time 0 = value of cell 2 at time 0 if value of cell 1 at time 0 = 2 and value of cell 2 at time 0 = 2, then value of cell 1 at time 0 = value of cell 2 at time 0
2 the PC at time (0 + 1) = (0 + 1) + 2 if instruction #0 is beq left=1 right=2 imm=2 and the PC at time 0 = 0 and value of cell 1 at time 0 = value of cell 2 at time 0, then the PC at time (0 + 1) = (0 + 1) + 2
3 0 + 1 = 1 0 + 1 = 1
4 (0 + 1) + 2 = 3 (0 + 1) + 2 = 3
5 the PC at time (0 + 1) = 3 if the PC at time (0 + 1) = (0 + 1) + 2 and (0 + 1) + 2 = 3, then the PC at time (0 + 1) = 3
6 the PC at time (0 + 1) = the PC at time 1 if 0 + 1 = 1, then the PC at time (0 + 1) = the PC at time 1
7 the PC at time 1 = 3 if the PC at time (0 + 1) = the PC at time 1 and the PC at time (0 + 1) = 3, then the PC at time 1 = 3

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