Proof: Do Control Map At Unchanged 15

Let's prove the following theorem:

if the following are true:

expression state at time 15 = "begin_expr"
Control Map at time 15 = [ entry 0: (pair ("while", 2)), [ ] ]

then Control Map at time 16 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof:

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Given

1	expression state at time 15 = "begin_expr"
2	Control Map at time 15 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof Table

#	Claim	Reason
1	Control Map at time (15 + 1) = Control Map at time 15	if expression state at time 15 = "begin_expr", then Control Map at time (15 + 1) = Control Map at time 15 (Insn Count Begin Expression (7))
2	Control Map at time (15 + 1) = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (15 + 1) = Control Map at time 15 and Control Map at time 15 = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time (15 + 1) = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality)
3	15 + 1 = 16	15 + 1 = 16 (Fixed Value Statement)
4	Control Map at time (15 + 1) = Control Map at time 16	if 15 + 1 = 16, then Control Map at time (15 + 1) = Control Map at time 16 (Substitution)
5	Control Map at time 16 = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (15 + 1) = Control Map at time 16 and Control Map at time (15 + 1) = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time 16 = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality Variation 2)

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