Proof: Do Control Map At Unchanged 30

Let's prove the following theorem:

if the following are true:

expression state at time 30 = "begin_expr"
Control Map at time 30 = [ entry 0: (pair ("while", 2)), [ ] ]

then Control Map at time 31 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof:

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Given

1	expression state at time 30 = "begin_expr"
2	Control Map at time 30 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof Table

#	Claim	Reason
1	Control Map at time (30 + 1) = Control Map at time 30	if expression state at time 30 = "begin_expr", then Control Map at time (30 + 1) = Control Map at time 30 (Insn Count Begin Expression (7))
2	Control Map at time (30 + 1) = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (30 + 1) = Control Map at time 30 and Control Map at time 30 = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time (30 + 1) = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality)
3	30 + 1 = 31	30 + 1 = 31 (Fixed Value Statement)
4	Control Map at time (30 + 1) = Control Map at time 31	if 30 + 1 = 31, then Control Map at time (30 + 1) = Control Map at time 31 (Substitution)
5	Control Map at time 31 = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (30 + 1) = Control Map at time 31 and Control Map at time (30 + 1) = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time 31 = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality Variation 2)

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