Proof: Do Control Map At Unchanged 20

Let's prove the following theorem:

if the following are true:

expression state at time 20 = "return"
Control Map at time 20 = [ entry 0: (pair ("function", "add_three")), [ ] ]

then Control Map at time 21 = [ entry 0: (pair ("function", "add_three")), [ ] ]

Proof:

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Given

1	expression state at time 20 = "return"
2	Control Map at time 20 = [ entry 0: (pair ("function", "add_three")), [ ] ]

Proof Table

#	Claim	Reason
1	Control Map at time (20 + 1) = Control Map at time 20	if expression state at time 20 = "return", then Control Map at time (20 + 1) = Control Map at time 20 (Return (10))
2	Control Map at time (20 + 1) = [ entry 0: (pair ("function", "add_three")), [ ] ]	if Control Map at time (20 + 1) = Control Map at time 20 and Control Map at time 20 = [ entry 0: (pair ("function", "add_three")), [ ] ], then Control Map at time (20 + 1) = [ entry 0: (pair ("function", "add_three")), [ ] ] (Transitive Property of Equality)
3	20 + 1 = 21	20 + 1 = 21 (Fixed Value Statement)
4	Control Map at time (20 + 1) = Control Map at time 21	if 20 + 1 = 21, then Control Map at time (20 + 1) = Control Map at time 21 (Substitution)
5	Control Map at time 21 = [ entry 0: (pair ("function", "add_three")), [ ] ]	if Control Map at time (20 + 1) = Control Map at time 21 and Control Map at time (20 + 1) = [ entry 0: (pair ("function", "add_three")), [ ] ], then Control Map at time 21 = [ entry 0: (pair ("function", "add_three")), [ ] ] (Transitive Property of Equality Variation 2)

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