Proof: Do Control Map At Unchanged 19

Let's prove the following theorem:

if the following are true:

expression state at time 19 = "return"
Control Map at time 19 = [ entry 0: (pair ("while", 2)), [ ] ]

then Control Map at time 20 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof:

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Given

1	expression state at time 19 = "return"
2	Control Map at time 19 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof Table

#	Claim	Reason
1	Control Map at time (19 + 1) = Control Map at time 19	if expression state at time 19 = "return", then Control Map at time (19 + 1) = Control Map at time 19 (Return (10))
2	Control Map at time (19 + 1) = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (19 + 1) = Control Map at time 19 and Control Map at time 19 = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time (19 + 1) = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality)
3	19 + 1 = 20	19 + 1 = 20 (Fixed Value Statement)
4	Control Map at time (19 + 1) = Control Map at time 20	if 19 + 1 = 20, then Control Map at time (19 + 1) = Control Map at time 20 (Substitution)
5	Control Map at time 20 = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (19 + 1) = Control Map at time 20 and Control Map at time (19 + 1) = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time 20 = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality Variation 2)

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