Proof: Proof: A Program that Computes Fibonacci Numbers

In this example, we present a program that computes Fibonacci numbers.

The Fibonacci numbers are a sequence where every number is the sum of the two previous numbers. Here are the first 6:

0, 1, 1, 2, 3, 5

Then we show that

Let's prove the following theorem:

if the following are true:

the PC at time 0 = 0
instruction #0 is addi dst=1 src=0 imm=0
instruction #1 is addi dst=2 src=0 imm=1
instruction #2 is addi dst=4 src=0 imm=0
instruction #3 is addi dst=5 src=0 imm=4
instruction #4 is add dst=3 src1=1 src2=2
instruction #5 is addi dst=1 src=2 imm=0
instruction #6 is addi dst=2 src=3 imm=0
instruction #7 is addi dst=4 src=4 imm=1
instruction #8 is beq left=4 right=5 imm=1
instruction #9 is jump imm=4

then value of cell 3 at time 25 = 5

After the following program is executed, cell #3 will contain the 6th Fibonacci number (5).

We need at least 2 numbers before we can calculate sums, so we start with 0 and 1. But notice that 5 is the sum of the two previous numbers 2 and 3.

Instructions

Memory Cells

Program Counter	Time
0	0

LW Computer Simulator

Memory cell #1 contains the nth Fibonacci number (initially the 1st Fibonacci number 0).

Cell #2 contains the n + 1th Fibonacci number (initially the 2nd Fibonacci number 1).

Memory cell #4 contains the current count or index.

Cell #5 contains 4. In each iteration, the computer will increment #4 and finish when it reaches the value in cell #5.

The computer computes the next Fibonacci number by adding values in cells #1 and #2 and storing it in cell #3. Then, to prepare for the next iteration, the values are "moved down" 1: cell #2 is copied to cell #1 and cell #3 is copied to cell #2.

Then the counter increments by 1, and the computer jumps back to instruction #4, and the next iteration begins.

At the end of the program, cell #3 contains the 6th Fibonacci number.

Proof:

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Given

1	the PC at time 0 = 0
2	instruction #0 is `addi dst=1 src=0 imm=0`
3	instruction #1 is `addi dst=2 src=0 imm=1`
4	instruction #2 is `addi dst=4 src=0 imm=0`
5	instruction #3 is `addi dst=5 src=0 imm=4`
6	instruction #4 is `add dst=3 src1=1 src2=2`
7	instruction #5 is `addi dst=1 src=2 imm=0`
8	instruction #6 is `addi dst=2 src=3 imm=0`
9	instruction #7 is `addi dst=4 src=4 imm=1`
10	instruction #8 is `beq left=4 right=5 imm=1`
11	instruction #9 is `jump imm=4`

Proof Table

#	Claim	Reason
1	value of cell 1 at time 1 = 0	if instruction #0 is `addi dst=1 src=0 imm=0` and the PC at time 0 = 0, then value of cell 1 at time 1 = 0 (Addi Zero Insn 0)
2	the PC at time 1 = 1	if instruction #0 is `addi dst=1 src=0 imm=0` and the PC at time 0 = 0, then the PC at time 1 = 1 (Pc 0)
3	value of cell 2 at time 2 = 1	if instruction #1 is `addi dst=2 src=0 imm=1` and the PC at time 1 = 1, then value of cell 2 at time 2 = 1 (Addi Zero Insn 1)
4	the PC at time 2 = 2	if instruction #1 is `addi dst=2 src=0 imm=1` and the PC at time 1 = 1, then the PC at time 2 = 2 (Pc 1)
5	value of cell 1 at time 2 = 0	if instruction #1 is `addi dst=2 src=0 imm=1` and the PC at time 1 = 1 and value of cell 1 at time 1 = 0, then value of cell 1 at time 2 = 0 (Byte 1 Stays the Same 1)
6	value of cell 4 at time 3 = 0	if instruction #2 is `addi dst=4 src=0 imm=0` and the PC at time 2 = 2, then value of cell 4 at time 3 = 0 (Addi Zero Insn 2)
7	the PC at time 3 = 3	if instruction #2 is `addi dst=4 src=0 imm=0` and the PC at time 2 = 2, then the PC at time 3 = 3 (Pc 2)
8	value of cell 1 at time 3 = 0	if instruction #2 is `addi dst=4 src=0 imm=0` and the PC at time 2 = 2 and value of cell 1 at time 2 = 0, then value of cell 1 at time 3 = 0 (Byte 1 Stays the Same 2)
9	value of cell 2 at time 3 = 1	if instruction #2 is `addi dst=4 src=0 imm=0` and the PC at time 2 = 2 and value of cell 2 at time 2 = 1, then value of cell 2 at time 3 = 1 (Byte 2 Stays the Same 2)
10	value of cell 5 at time 4 = 4	if instruction #3 is `addi dst=5 src=0 imm=4` and the PC at time 3 = 3, then value of cell 5 at time 4 = 4 (Addi Zero Insn 3)
11	the PC at time 4 = 4	if instruction #3 is `addi dst=5 src=0 imm=4` and the PC at time 3 = 3, then the PC at time 4 = 4 (Pc 3)
12	value of cell 1 at time 4 = 0	if instruction #3 is `addi dst=5 src=0 imm=4` and the PC at time 3 = 3 and value of cell 1 at time 3 = 0, then value of cell 1 at time 4 = 0 (Byte 1 Stays the Same 3)
13	value of cell 2 at time 4 = 1	if instruction #3 is `addi dst=5 src=0 imm=4` and the PC at time 3 = 3 and value of cell 2 at time 3 = 1, then value of cell 2 at time 4 = 1 (Byte 2 Stays the Same 3)
14	value of cell 4 at time 4 = 0	if instruction #3 is `addi dst=5 src=0 imm=4` and the PC at time 3 = 3 and value of cell 4 at time 3 = 0, then value of cell 4 at time 4 = 0 (Byte 4 Stays the Same 3)
15	value of cell 3 at time 5 = (value of cell 1 at time 4) + (value of cell 2 at time 4)	if the PC at time 4 = 4 and instruction #4 is `add dst=3 src1=1 src2=2`, then value of cell 3 at time 5 = (value of cell 1 at time 4) + (value of cell 2 at time 4) (Add Insn 4)
16	value of cell 3 at time 5 = 1	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 = 0 and value of cell 2 at time 4 = 1, then value of cell 3 at time 5 = 1 (Substitute Add 4)
17	the PC at time 5 = 5	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 4 = 4, then the PC at time 5 = 5 (Pc 4)
18	value of cell 1 at time 5 = 0	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 4 = 4 and value of cell 1 at time 4 = 0, then value of cell 1 at time 5 = 0 (Byte 1 Stays the Same 4)
19	value of cell 2 at time 5 = 1	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 4 = 4 and value of cell 2 at time 4 = 1, then value of cell 2 at time 5 = 1 (Byte 2 Stays the Same 4)
20	value of cell 4 at time 5 = 0	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 4 = 4 and value of cell 4 at time 4 = 0, then value of cell 4 at time 5 = 0 (Byte 4 Stays the Same 4)
21	value of cell 5 at time 5 = 4	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 4 = 4 and value of cell 5 at time 4 = 4, then value of cell 5 at time 5 = 4 (Byte 5 Stays the Same 4)
22	value of cell 1 at time 6 = 1	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 5 = 5 and value of cell 2 at time 5 = 1, then value of cell 1 at time 6 = 1 (Addi Insn 5)
23	the PC at time 6 = 6	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 5 = 5, then the PC at time 6 = 6 (Pc 5)
24	value of cell 2 at time 6 = 1	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 5 = 5 and value of cell 2 at time 5 = 1, then value of cell 2 at time 6 = 1 (Byte 2 Stays the Same 5)
25	value of cell 4 at time 6 = 0	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 5 = 5 and value of cell 4 at time 5 = 0, then value of cell 4 at time 6 = 0 (Byte 4 Stays the Same 5)
26	value of cell 5 at time 6 = 4	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 5 = 5 and value of cell 5 at time 5 = 4, then value of cell 5 at time 6 = 4 (Byte 5 Stays the Same 5)
27	value of cell 3 at time 6 = 1	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 5 = 5 and value of cell 3 at time 5 = 1, then value of cell 3 at time 6 = 1 (Byte 3 Stays the Same 5)
28	value of cell 2 at time 7 = 1	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 6 = 6 and value of cell 3 at time 6 = 1, then value of cell 2 at time 7 = 1 (Addi Insn 6)
29	the PC at time 7 = 7	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 6 = 6, then the PC at time 7 = 7 (Pc 6)
30	value of cell 1 at time 7 = 1	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 6 = 6 and value of cell 1 at time 6 = 1, then value of cell 1 at time 7 = 1 (Byte 1 Stays the Same 6)
31	value of cell 4 at time 7 = 0	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 6 = 6 and value of cell 4 at time 6 = 0, then value of cell 4 at time 7 = 0 (Byte 4 Stays the Same 6)
32	value of cell 5 at time 7 = 4	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 6 = 6 and value of cell 5 at time 6 = 4, then value of cell 5 at time 7 = 4 (Byte 5 Stays the Same 6)
33	value of cell 3 at time 7 = 1	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 6 = 6 and value of cell 3 at time 6 = 1, then value of cell 3 at time 7 = 1 (Byte 3 Stays the Same 6)
34	value of cell 4 at time 8 = 1	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 7 = 7 and value of cell 4 at time 7 = 0, then value of cell 4 at time 8 = 1 (Addi Insn 7)
35	the PC at time 8 = 8	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 7 = 7, then the PC at time 8 = 8 (Pc 7)
36	value of cell 1 at time 8 = 1	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 7 = 7 and value of cell 1 at time 7 = 1, then value of cell 1 at time 8 = 1 (Byte 1 Stays the Same 7)
37	value of cell 2 at time 8 = 1	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 7 = 7 and value of cell 2 at time 7 = 1, then value of cell 2 at time 8 = 1 (Byte 2 Stays the Same 7)
38	value of cell 5 at time 8 = 4	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 7 = 7 and value of cell 5 at time 7 = 4, then value of cell 5 at time 8 = 4 (Byte 5 Stays the Same 7)
39	value of cell 3 at time 8 = 1	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 7 = 7 and value of cell 3 at time 7 = 1, then value of cell 3 at time 8 = 1 (Byte 3 Stays the Same 7)
40	the PC at time 9 = 9	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 8 = 8 and value of cell 4 at time 8 = 1 and value of cell 5 at time 8 = 4, then the PC at time 9 = 9 (Beq No Branch 8)
41	value of cell 1 at time 9 = 1	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 8 = 8 and value of cell 1 at time 8 = 1, then value of cell 1 at time 9 = 1 (Byte 1 Stays the Same 8)
42	value of cell 2 at time 9 = 1	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 8 = 8 and value of cell 2 at time 8 = 1, then value of cell 2 at time 9 = 1 (Byte 2 Stays the Same 8)
43	value of cell 4 at time 9 = 1	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 8 = 8 and value of cell 4 at time 8 = 1, then value of cell 4 at time 9 = 1 (Byte 4 Stays the Same 8)
44	value of cell 5 at time 9 = 4	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 8 = 8 and value of cell 5 at time 8 = 4, then value of cell 5 at time 9 = 4 (Byte 5 Stays the Same 8)
45	value of cell 3 at time 9 = 1	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 8 = 8 and value of cell 3 at time 8 = 1, then value of cell 3 at time 9 = 1 (Byte 3 Stays the Same 8)
46	the PC at time 10 = 4	if instruction #9 is `jump imm=4` and the PC at time 9 = 9, then the PC at time 10 = 4 (Jump Help 9)
47	value of cell 1 at time 10 = 1	if instruction #9 is `jump imm=4` and the PC at time 9 = 9 and value of cell 1 at time 9 = 1, then value of cell 1 at time 10 = 1 (Byte 1 Stays the Same 9)
48	value of cell 2 at time 10 = 1	if instruction #9 is `jump imm=4` and the PC at time 9 = 9 and value of cell 2 at time 9 = 1, then value of cell 2 at time 10 = 1 (Byte 2 Stays the Same 9)
49	value of cell 4 at time 10 = 1	if instruction #9 is `jump imm=4` and the PC at time 9 = 9 and value of cell 4 at time 9 = 1, then value of cell 4 at time 10 = 1 (Byte 4 Stays the Same 9)
50	value of cell 5 at time 10 = 4	if instruction #9 is `jump imm=4` and the PC at time 9 = 9 and value of cell 5 at time 9 = 4, then value of cell 5 at time 10 = 4 (Byte 5 Stays the Same 9)
51	value of cell 3 at time 10 = 1	if instruction #9 is `jump imm=4` and the PC at time 9 = 9 and value of cell 3 at time 9 = 1, then value of cell 3 at time 10 = 1 (Byte 3 Stays the Same 9)
52	value of cell 3 at time 11 = (value of cell 1 at time 10) + (value of cell 2 at time 10)	if the PC at time 10 = 4 and instruction #4 is `add dst=3 src1=1 src2=2`, then value of cell 3 at time 11 = (value of cell 1 at time 10) + (value of cell 2 at time 10) (Add Insn 10)
53	value of cell 3 at time 11 = 2	if value of cell 3 at time 11 = (value of cell 1 at time 10) + (value of cell 2 at time 10) and value of cell 1 at time 10 = 1 and value of cell 2 at time 10 = 1, then value of cell 3 at time 11 = 2 (Substitute Add 10)
54	the PC at time 11 = 5	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 10 = 4, then the PC at time 11 = 5 (Pc 10)
55	value of cell 1 at time 11 = 1	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 10 = 4 and value of cell 1 at time 10 = 1, then value of cell 1 at time 11 = 1 (Byte 1 Stays the Same 10)
56	value of cell 2 at time 11 = 1	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 10 = 4 and value of cell 2 at time 10 = 1, then value of cell 2 at time 11 = 1 (Byte 2 Stays the Same 10)
57	value of cell 4 at time 11 = 1	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 10 = 4 and value of cell 4 at time 10 = 1, then value of cell 4 at time 11 = 1 (Byte 4 Stays the Same 10)
58	value of cell 5 at time 11 = 4	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 10 = 4 and value of cell 5 at time 10 = 4, then value of cell 5 at time 11 = 4 (Byte 5 Stays the Same 10)
59	value of cell 1 at time 12 = 1	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 11 = 5 and value of cell 2 at time 11 = 1, then value of cell 1 at time 12 = 1 (Addi Insn 11)
60	the PC at time 12 = 6	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 11 = 5, then the PC at time 12 = 6 (Pc 11)
61	value of cell 2 at time 12 = 1	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 11 = 5 and value of cell 2 at time 11 = 1, then value of cell 2 at time 12 = 1 (Byte 2 Stays the Same 11)
62	value of cell 4 at time 12 = 1	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 11 = 5 and value of cell 4 at time 11 = 1, then value of cell 4 at time 12 = 1 (Byte 4 Stays the Same 11)
63	value of cell 5 at time 12 = 4	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 11 = 5 and value of cell 5 at time 11 = 4, then value of cell 5 at time 12 = 4 (Byte 5 Stays the Same 11)
64	value of cell 3 at time 12 = 2	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 11 = 5 and value of cell 3 at time 11 = 2, then value of cell 3 at time 12 = 2 (Byte 3 Stays the Same 11)
65	value of cell 2 at time 13 = 2	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 12 = 6 and value of cell 3 at time 12 = 2, then value of cell 2 at time 13 = 2 (Addi Insn 12)
66	the PC at time 13 = 7	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 12 = 6, then the PC at time 13 = 7 (Pc 12)
67	value of cell 1 at time 13 = 1	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 12 = 6 and value of cell 1 at time 12 = 1, then value of cell 1 at time 13 = 1 (Byte 1 Stays the Same 12)
68	value of cell 4 at time 13 = 1	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 12 = 6 and value of cell 4 at time 12 = 1, then value of cell 4 at time 13 = 1 (Byte 4 Stays the Same 12)
69	value of cell 5 at time 13 = 4	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 12 = 6 and value of cell 5 at time 12 = 4, then value of cell 5 at time 13 = 4 (Byte 5 Stays the Same 12)
70	value of cell 3 at time 13 = 2	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 12 = 6 and value of cell 3 at time 12 = 2, then value of cell 3 at time 13 = 2 (Byte 3 Stays the Same 12)
71	value of cell 4 at time 14 = 2	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 13 = 7 and value of cell 4 at time 13 = 1, then value of cell 4 at time 14 = 2 (Addi Insn 13)
72	the PC at time 14 = 8	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 13 = 7, then the PC at time 14 = 8 (Pc 13)
73	value of cell 1 at time 14 = 1	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 13 = 7 and value of cell 1 at time 13 = 1, then value of cell 1 at time 14 = 1 (Byte 1 Stays the Same 13)
74	value of cell 2 at time 14 = 2	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 13 = 7 and value of cell 2 at time 13 = 2, then value of cell 2 at time 14 = 2 (Byte 2 Stays the Same 13)
75	value of cell 5 at time 14 = 4	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 13 = 7 and value of cell 5 at time 13 = 4, then value of cell 5 at time 14 = 4 (Byte 5 Stays the Same 13)
76	value of cell 3 at time 14 = 2	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 13 = 7 and value of cell 3 at time 13 = 2, then value of cell 3 at time 14 = 2 (Byte 3 Stays the Same 13)
77	the PC at time 15 = 9	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 14 = 8 and value of cell 4 at time 14 = 2 and value of cell 5 at time 14 = 4, then the PC at time 15 = 9 (Beq No Branch 14)
78	value of cell 1 at time 15 = 1	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 14 = 8 and value of cell 1 at time 14 = 1, then value of cell 1 at time 15 = 1 (Byte 1 Stays the Same 14)
79	value of cell 2 at time 15 = 2	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 14 = 8 and value of cell 2 at time 14 = 2, then value of cell 2 at time 15 = 2 (Byte 2 Stays the Same 14)
80	value of cell 4 at time 15 = 2	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 14 = 8 and value of cell 4 at time 14 = 2, then value of cell 4 at time 15 = 2 (Byte 4 Stays the Same 14)
81	value of cell 5 at time 15 = 4	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 14 = 8 and value of cell 5 at time 14 = 4, then value of cell 5 at time 15 = 4 (Byte 5 Stays the Same 14)
82	value of cell 3 at time 15 = 2	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 14 = 8 and value of cell 3 at time 14 = 2, then value of cell 3 at time 15 = 2 (Byte 3 Stays the Same 14)
83	the PC at time 16 = 4	if instruction #9 is `jump imm=4` and the PC at time 15 = 9, then the PC at time 16 = 4 (Jump Help 15)
84	value of cell 1 at time 16 = 1	if instruction #9 is `jump imm=4` and the PC at time 15 = 9 and value of cell 1 at time 15 = 1, then value of cell 1 at time 16 = 1 (Byte 1 Stays the Same 15)
85	value of cell 2 at time 16 = 2	if instruction #9 is `jump imm=4` and the PC at time 15 = 9 and value of cell 2 at time 15 = 2, then value of cell 2 at time 16 = 2 (Byte 2 Stays the Same 15)
86	value of cell 4 at time 16 = 2	if instruction #9 is `jump imm=4` and the PC at time 15 = 9 and value of cell 4 at time 15 = 2, then value of cell 4 at time 16 = 2 (Byte 4 Stays the Same 15)
87	value of cell 5 at time 16 = 4	if instruction #9 is `jump imm=4` and the PC at time 15 = 9 and value of cell 5 at time 15 = 4, then value of cell 5 at time 16 = 4 (Byte 5 Stays the Same 15)
88	value of cell 3 at time 16 = 2	if instruction #9 is `jump imm=4` and the PC at time 15 = 9 and value of cell 3 at time 15 = 2, then value of cell 3 at time 16 = 2 (Byte 3 Stays the Same 15)
89	value of cell 3 at time 17 = (value of cell 1 at time 16) + (value of cell 2 at time 16)	if the PC at time 16 = 4 and instruction #4 is `add dst=3 src1=1 src2=2`, then value of cell 3 at time 17 = (value of cell 1 at time 16) + (value of cell 2 at time 16) (Add Insn 16)
90	value of cell 3 at time 17 = 3	if value of cell 3 at time 17 = (value of cell 1 at time 16) + (value of cell 2 at time 16) and value of cell 1 at time 16 = 1 and value of cell 2 at time 16 = 2, then value of cell 3 at time 17 = 3 (Substitute Add 16)
91	the PC at time 17 = 5	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 16 = 4, then the PC at time 17 = 5 (Pc 16)
92	value of cell 1 at time 17 = 1	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 16 = 4 and value of cell 1 at time 16 = 1, then value of cell 1 at time 17 = 1 (Byte 1 Stays the Same 16)
93	value of cell 2 at time 17 = 2	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 16 = 4 and value of cell 2 at time 16 = 2, then value of cell 2 at time 17 = 2 (Byte 2 Stays the Same 16)
94	value of cell 4 at time 17 = 2	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 16 = 4 and value of cell 4 at time 16 = 2, then value of cell 4 at time 17 = 2 (Byte 4 Stays the Same 16)
95	value of cell 5 at time 17 = 4	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 16 = 4 and value of cell 5 at time 16 = 4, then value of cell 5 at time 17 = 4 (Byte 5 Stays the Same 16)
96	value of cell 1 at time 18 = 2	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 17 = 5 and value of cell 2 at time 17 = 2, then value of cell 1 at time 18 = 2 (Addi Insn 17)
97	the PC at time 18 = 6	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 17 = 5, then the PC at time 18 = 6 (Pc 17)
98	value of cell 2 at time 18 = 2	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 17 = 5 and value of cell 2 at time 17 = 2, then value of cell 2 at time 18 = 2 (Byte 2 Stays the Same 17)
99	value of cell 4 at time 18 = 2	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 17 = 5 and value of cell 4 at time 17 = 2, then value of cell 4 at time 18 = 2 (Byte 4 Stays the Same 17)
100	value of cell 5 at time 18 = 4	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 17 = 5 and value of cell 5 at time 17 = 4, then value of cell 5 at time 18 = 4 (Byte 5 Stays the Same 17)
101	value of cell 3 at time 18 = 3	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 17 = 5 and value of cell 3 at time 17 = 3, then value of cell 3 at time 18 = 3 (Byte 3 Stays the Same 17)
102	value of cell 2 at time 19 = 3	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 18 = 6 and value of cell 3 at time 18 = 3, then value of cell 2 at time 19 = 3 (Addi Insn 18)
103	the PC at time 19 = 7	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 18 = 6, then the PC at time 19 = 7 (Pc 18)
104	value of cell 1 at time 19 = 2	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 18 = 6 and value of cell 1 at time 18 = 2, then value of cell 1 at time 19 = 2 (Byte 1 Stays the Same 18)
105	value of cell 4 at time 19 = 2	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 18 = 6 and value of cell 4 at time 18 = 2, then value of cell 4 at time 19 = 2 (Byte 4 Stays the Same 18)
106	value of cell 5 at time 19 = 4	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 18 = 6 and value of cell 5 at time 18 = 4, then value of cell 5 at time 19 = 4 (Byte 5 Stays the Same 18)
107	value of cell 3 at time 19 = 3	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 18 = 6 and value of cell 3 at time 18 = 3, then value of cell 3 at time 19 = 3 (Byte 3 Stays the Same 18)
108	value of cell 4 at time 20 = 3	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 19 = 7 and value of cell 4 at time 19 = 2, then value of cell 4 at time 20 = 3 (Addi Insn 19)
109	the PC at time 20 = 8	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 19 = 7, then the PC at time 20 = 8 (Pc 19)
110	value of cell 1 at time 20 = 2	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 19 = 7 and value of cell 1 at time 19 = 2, then value of cell 1 at time 20 = 2 (Byte 1 Stays the Same 19)
111	value of cell 2 at time 20 = 3	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 19 = 7 and value of cell 2 at time 19 = 3, then value of cell 2 at time 20 = 3 (Byte 2 Stays the Same 19)
112	value of cell 5 at time 20 = 4	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 19 = 7 and value of cell 5 at time 19 = 4, then value of cell 5 at time 20 = 4 (Byte 5 Stays the Same 19)
113	value of cell 3 at time 20 = 3	if instruction #7 is `addi dst=4 src=4 imm=1` and the PC at time 19 = 7 and value of cell 3 at time 19 = 3, then value of cell 3 at time 20 = 3 (Byte 3 Stays the Same 19)
114	the PC at time 21 = 9	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 20 = 8 and value of cell 4 at time 20 = 3 and value of cell 5 at time 20 = 4, then the PC at time 21 = 9 (Beq No Branch 20)
115	value of cell 1 at time 21 = 2	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 20 = 8 and value of cell 1 at time 20 = 2, then value of cell 1 at time 21 = 2 (Byte 1 Stays the Same 20)
116	value of cell 2 at time 21 = 3	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 20 = 8 and value of cell 2 at time 20 = 3, then value of cell 2 at time 21 = 3 (Byte 2 Stays the Same 20)
117	value of cell 4 at time 21 = 3	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 20 = 8 and value of cell 4 at time 20 = 3, then value of cell 4 at time 21 = 3 (Byte 4 Stays the Same 20)
118	value of cell 5 at time 21 = 4	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 20 = 8 and value of cell 5 at time 20 = 4, then value of cell 5 at time 21 = 4 (Byte 5 Stays the Same 20)
119	value of cell 3 at time 21 = 3	if instruction #8 is `beq left=4 right=5 imm=1` and the PC at time 20 = 8 and value of cell 3 at time 20 = 3, then value of cell 3 at time 21 = 3 (Byte 3 Stays the Same 20)
120	the PC at time 22 = 4	if instruction #9 is `jump imm=4` and the PC at time 21 = 9, then the PC at time 22 = 4 (Jump Help 21)
121	value of cell 1 at time 22 = 2	if instruction #9 is `jump imm=4` and the PC at time 21 = 9 and value of cell 1 at time 21 = 2, then value of cell 1 at time 22 = 2 (Byte 1 Stays the Same 21)
122	value of cell 2 at time 22 = 3	if instruction #9 is `jump imm=4` and the PC at time 21 = 9 and value of cell 2 at time 21 = 3, then value of cell 2 at time 22 = 3 (Byte 2 Stays the Same 21)
123	value of cell 4 at time 22 = 3	if instruction #9 is `jump imm=4` and the PC at time 21 = 9 and value of cell 4 at time 21 = 3, then value of cell 4 at time 22 = 3 (Byte 4 Stays the Same 21)
124	value of cell 5 at time 22 = 4	if instruction #9 is `jump imm=4` and the PC at time 21 = 9 and value of cell 5 at time 21 = 4, then value of cell 5 at time 22 = 4 (Byte 5 Stays the Same 21)
125	value of cell 3 at time 22 = 3	if instruction #9 is `jump imm=4` and the PC at time 21 = 9 and value of cell 3 at time 21 = 3, then value of cell 3 at time 22 = 3 (Byte 3 Stays the Same 21)
126	value of cell 3 at time 23 = (value of cell 1 at time 22) + (value of cell 2 at time 22)	if the PC at time 22 = 4 and instruction #4 is `add dst=3 src1=1 src2=2`, then value of cell 3 at time 23 = (value of cell 1 at time 22) + (value of cell 2 at time 22) (Add Insn 22)
127	value of cell 3 at time 23 = 5	if value of cell 3 at time 23 = (value of cell 1 at time 22) + (value of cell 2 at time 22) and value of cell 1 at time 22 = 2 and value of cell 2 at time 22 = 3, then value of cell 3 at time 23 = 5 (Substitute Add 22)
128	the PC at time 23 = 5	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 22 = 4, then the PC at time 23 = 5 (Pc 22)
129	value of cell 1 at time 23 = 2	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 22 = 4 and value of cell 1 at time 22 = 2, then value of cell 1 at time 23 = 2 (Byte 1 Stays the Same 22)
130	value of cell 2 at time 23 = 3	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 22 = 4 and value of cell 2 at time 22 = 3, then value of cell 2 at time 23 = 3 (Byte 2 Stays the Same 22)
131	value of cell 4 at time 23 = 3	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 22 = 4 and value of cell 4 at time 22 = 3, then value of cell 4 at time 23 = 3 (Byte 4 Stays the Same 22)
132	value of cell 5 at time 23 = 4	if instruction #4 is `add dst=3 src1=1 src2=2` and the PC at time 22 = 4 and value of cell 5 at time 22 = 4, then value of cell 5 at time 23 = 4 (Byte 5 Stays the Same 22)
133	value of cell 1 at time 24 = 3	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 23 = 5 and value of cell 2 at time 23 = 3, then value of cell 1 at time 24 = 3 (Addi Insn 23)
134	the PC at time 24 = 6	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 23 = 5, then the PC at time 24 = 6 (Pc 23)
135	value of cell 2 at time 24 = 3	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 23 = 5 and value of cell 2 at time 23 = 3, then value of cell 2 at time 24 = 3 (Byte 2 Stays the Same 23)
136	value of cell 4 at time 24 = 3	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 23 = 5 and value of cell 4 at time 23 = 3, then value of cell 4 at time 24 = 3 (Byte 4 Stays the Same 23)
137	value of cell 5 at time 24 = 4	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 23 = 5 and value of cell 5 at time 23 = 4, then value of cell 5 at time 24 = 4 (Byte 5 Stays the Same 23)
138	value of cell 3 at time 24 = 5	if instruction #5 is `addi dst=1 src=2 imm=0` and the PC at time 23 = 5 and value of cell 3 at time 23 = 5, then value of cell 3 at time 24 = 5 (Byte 3 Stays the Same 23)
139	value of cell 2 at time 25 = 5	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 24 = 6 and value of cell 3 at time 24 = 5, then value of cell 2 at time 25 = 5 (Addi Insn 24)
140	the PC at time 25 = 7	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 24 = 6, then the PC at time 25 = 7 (Pc 24)
141	value of cell 1 at time 25 = 3	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 24 = 6 and value of cell 1 at time 24 = 3, then value of cell 1 at time 25 = 3 (Byte 1 Stays the Same 24)
142	value of cell 4 at time 25 = 3	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 24 = 6 and value of cell 4 at time 24 = 3, then value of cell 4 at time 25 = 3 (Byte 4 Stays the Same 24)
143	value of cell 5 at time 25 = 4	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 24 = 6 and value of cell 5 at time 24 = 4, then value of cell 5 at time 25 = 4 (Byte 5 Stays the Same 24)
144	value of cell 3 at time 25 = 5	if instruction #6 is `addi dst=2 src=3 imm=0` and the PC at time 24 = 6 and value of cell 3 at time 24 = 5, then value of cell 3 at time 25 = 5 (Byte 3 Stays the Same 24)

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