Proof: Do Stack At Unchanged 20

Let's prove the following theorem:

if the following are true:
  • expression state at time 20 = "begin_expr"
  • stack at time 20 = [ program context with variables: [ ] and expression state: (expression state with parent stack: [ ] arguments: [ ] values: [ ] line: 10 tab: 0) and control map: [ entry 0: (pair ("class", "Dog")), [ ] ], [ ] ]

then stack at time 21 = [ program context with variables: [ ] and expression state: (expression state with parent stack: [ ] arguments: [ ] values: [ ] line: 10 tab: 0) and control map: [ entry 0: (pair ("class", "Dog")), [ ] ], [ ] ]

Proof:

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Given
1 expression state at time 20 = "begin_expr"
2 stack at time 20 = [ program context with variables: [ ] and expression state: (expression state with parent stack: [ ] arguments: [ ] values: [ ] line: 10 tab: 0) and control map: [ entry 0: (pair ("class", "Dog")), [ ] ], [ ] ]
Proof Table
# Claim Reason
1 stack at time (20 + 1) = stack at time 20 if expression state at time 20 = "begin_expr", then stack at time (20 + 1) = stack at time 20
2 stack at time (20 + 1) = [ program context with variables: [ ] and expression state: (expression state with parent stack: [ ] arguments: [ ] values: [ ] line: 10 tab: 0) and control map: [ entry 0: (pair ("class", "Dog")), [ ] ], [ ] ] if stack at time (20 + 1) = stack at time 20 and stack at time 20 = [ program context with variables: [ ] and expression state: (expression state with parent stack: [ ] arguments: [ ] values: [ ] line: 10 tab: 0) and control map: [ entry 0: (pair ("class", "Dog")), [ ] ], [ ] ], then stack at time (20 + 1) = [ program context with variables: [ ] and expression state: (expression state with parent stack: [ ] arguments: [ ] values: [ ] line: 10 tab: 0) and control map: [ entry 0: (pair ("class", "Dog")), [ ] ], [ ] ]
3 20 + 1 = 21 20 + 1 = 21
4 stack at time (20 + 1) = stack at time 21 if 20 + 1 = 21, then stack at time (20 + 1) = stack at time 21
5 stack at time 21 = [ program context with variables: [ ] and expression state: (expression state with parent stack: [ ] arguments: [ ] values: [ ] line: 10 tab: 0) and control map: [ entry 0: (pair ("class", "Dog")), [ ] ], [ ] ] if stack at time (20 + 1) = stack at time 21 and stack at time (20 + 1) = [ program context with variables: [ ] and expression state: (expression state with parent stack: [ ] arguments: [ ] values: [ ] line: 10 tab: 0) and control map: [ entry 0: (pair ("class", "Dog")), [ ] ], [ ] ], then stack at time 21 = [ program context with variables: [ ] and expression state: (expression state with parent stack: [ ] arguments: [ ] values: [ ] line: 10 tab: 0) and control map: [ entry 0: (pair ("class", "Dog")), [ ] ], [ ] ]

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