Proof: Beq No Branch 8

Let's prove the following theorem:

if the following are true:

instruction #8 is beq left=4 right=5 imm=1
the PC at time 8 = 8
value of cell 4 at time 8 = 1
value of cell 5 at time 8 = 4

then the PC at time 9 = 9

Instructions

Memory Cells

Program Counter	Time
0	0

LW Computer Simulator

Proof:

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Given

1	instruction #8 is `beq left=4 right=5 imm=1`
2	the PC at time 8 = 8
3	value of cell 4 at time 8 = 1
4	value of cell 5 at time 8 = 4

Proof Table

#	Claim	Reason
1	value of cell 4 at time 8 = 4 = 1 = 4	if value of cell 4 at time 8 = 1, then value of cell 4 at time 8 = 4 = 1 = 4 (Substitution)
2	not (value of cell 4 at time 8 = 4) = not (1 = 4)	if value of cell 4 at time 8 = 4 = 1 = 4, then not (value of cell 4 at time 8 = 4) = not (1 = 4) (Substitution)
3	not (1 = 4)	not (1 = 4) (Fixed Value Statement)
4	not (value of cell 4 at time 8 = 4)	if not (1 = 4) and not (value of cell 4 at time 8 = 4) = not (1 = 4), then not (value of cell 4 at time 8 = 4) (Truth Propagation Property 2)
5	value of cell 4 at time 8 = value of cell 5 at time 8 = value of cell 4 at time 8 = 4	if value of cell 5 at time 8 = 4, then value of cell 4 at time 8 = value of cell 5 at time 8 = value of cell 4 at time 8 = 4 (Substitution)
6	not (value of cell 4 at time 8 = value of cell 5 at time 8) = not (value of cell 4 at time 8 = 4)	if value of cell 4 at time 8 = value of cell 5 at time 8 = value of cell 4 at time 8 = 4, then not (value of cell 4 at time 8 = value of cell 5 at time 8) = not (value of cell 4 at time 8 = 4) (Substitution)
7	not (value of cell 4 at time 8 = value of cell 5 at time 8)	if not (value of cell 4 at time 8 = 4) and not (value of cell 4 at time 8 = value of cell 5 at time 8) = not (value of cell 4 at time 8 = 4), then not (value of cell 4 at time 8 = value of cell 5 at time 8) (Truth Propagation Property 2)
8	the PC at time (8 + 1) = 8 + 1	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 8 = 8 and not (value of cell 4 at time 8 = value of cell 5 at time 8), then the PC at time (8 + 1) = 8 + 1 (BEQ Instruction Property 2)
9	8 + 1 = 9	8 + 1 = 9 (Fixed Value Statement)
10	the PC at time (8 + 1) = the PC at time 9	if 8 + 1 = 9, then the PC at time (8 + 1) = the PC at time 9 (Substitution)
11	the PC at time 9 = 9	if the PC at time (8 + 1) = 8 + 1 and the PC at time (8 + 1) = the PC at time 9 and 8 + 1 = 9, then the PC at time 9 = 9 (Transitive Property Application 2)

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