Proof: Do Control Map At Unchanged 31

Let's prove the following theorem:

if the following are true:

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

then Control Map at time 32 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof:

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Given

1	expression state at time 31 = "return"
2	Control Map at time 31 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof Table

#	Claim	Reason
1	Control Map at time (31 + 1) = Control Map at time 31	if expression state at time 31 = "return", then Control Map at time (31 + 1) = Control Map at time 31 (Return (10))
2	Control Map at time (31 + 1) = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (31 + 1) = Control Map at time 31 and Control Map at time 31 = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time (31 + 1) = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality)
3	31 + 1 = 32	31 + 1 = 32 (Fixed Value Statement)
4	Control Map at time (31 + 1) = Control Map at time 32	if 31 + 1 = 32, then Control Map at time (31 + 1) = Control Map at time 32 (Substitution)
5	Control Map at time 32 = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (31 + 1) = Control Map at time 32 and Control Map at time (31 + 1) = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time 32 = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality Variation 2)

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