Proof: Do Control Map At Unchanged 21

Let's prove the following theorem:

if the following are true:

expression state at time 21 = "call_function_begin"
Control Map at time 21 = [ entry 0: (pair ("while", 2)), [ ] ]

then Control Map at time 22 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof:

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Given

1	expression state at time 21 = "call_function_begin"
2	Control Map at time 21 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof Table

#	Claim	Reason
1	Control Map at time (21 + 1) = Control Map at time 21	if expression state at time 21 = "call_function_begin", then Control Map at time (21 + 1) = Control Map at time 21 (Assign Parent (9))
2	Control Map at time (21 + 1) = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (21 + 1) = Control Map at time 21 and Control Map at time 21 = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time (21 + 1) = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality)
3	21 + 1 = 22	21 + 1 = 22 (Fixed Value Statement)
4	Control Map at time (21 + 1) = Control Map at time 22	if 21 + 1 = 22, then Control Map at time (21 + 1) = Control Map at time 22 (Substitution)
5	Control Map at time 22 = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (21 + 1) = Control Map at time 22 and Control Map at time (21 + 1) = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time 22 = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality Variation 2)

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