# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 6
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 10 − i4_0 ≤ 0 ∧ i7_post ≤ 0 ∧ − i7_post ≤ 0 ∧ i7_0 − i7_post ≤ 0 ∧ − i7_0 + i7_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 0 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + i4_0 ≤ 0 ∧ −1 − i4_0 + i4_post ≤ 0 ∧ 1 + i4_0 − i4_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 2 2 0: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 1 3 3: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 3 4 4: 10 − i7_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 3 5 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + i7_0 ≤ 0 ∧ −1 − i7_0 + i7_post ≤ 0 ∧ 1 + i7_0 − i7_post ≤ 0 ∧ i7_0 − i7_post ≤ 0 ∧ − i7_0 + i7_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 5 6 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i4_post ≤ 0 ∧ − i4_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 6 7 5: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0

## Proof

The following invariants are asserted.

 0: TRUE 1: 10 − i4_0 ≤ 0 2: TRUE 3: 10 − i4_0 ≤ 0 4: 10 − i4_0 ≤ 0 ∧ 10 − i7_0 ≤ 0 5: TRUE 6: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) 10 − i4_0 ≤ 0 2 (2) TRUE 3 (3) 10 − i4_0 ≤ 0 4 (4) 10 − i4_0 ≤ 0 ∧ 10 − i7_0 ≤ 0 5 (5) TRUE 6 (6) TRUE
• initial node: 6
• cover edges:
• transition edges:  0 0 1 0 1 2 1 3 3 2 2 0 3 4 4 3 5 1 5 6 2 6 7 5

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 8 1: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 2 15 2: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i7_post + i7_post ≤ 0 ∧ i7_post − i7_post ≤ 0 ∧ − i7_0 + i7_0 ≤ 0 ∧ i7_0 − i7_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 4, 6, 7 using the following ranking functions, which are bounded by −17.

 6: 0 5: 0 0: 0 2: 0 1: 0 3: 0 4: 0 6: −6 5: −7 0: −8 2: −8 2_var_snapshot: −8 2*: −8 1: −9 3: −9 1_var_snapshot: −9 1*: −9 4: −10
Hints:
 9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 0 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 4 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 6 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 7 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

1* 11 1: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

1 9 1_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

2* 18 2: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

2 16 2_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0i7_post + i7_post ≤ 0i7_posti7_post ≤ 0i7_0 + i7_0 ≤ 0i7_0i7_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0

### 8 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

### 8.1 SCC Subproblem 1/2

Here we consider the SCC { 1, 3, 1_var_snapshot, 1* }.

### 8.1.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −30.

 1: −3⋅i7_0 3: −2 − 3⋅i7_0 1_var_snapshot: −1 − 3⋅i7_0 1*: −3⋅i7_0
Hints:
 9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] ] 11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] ] 3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] ] 5 lexStrict[ [0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 8.1.2 Transition Removal

We remove transitions 9, 11, 3 using the following ranking functions, which are bounded by −1.

 1: i4_0 3: − i4_0 1_var_snapshot: 0 1*: 1 + i4_0
Hints:
 9 lexStrict[ [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 11 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 3 lexStrict[ [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 8.1.3 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

### 8.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

### 8.1.3.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

### 8.2 SCC Subproblem 2/2

Here we consider the SCC { 0, 2, 2_var_snapshot, 2* }.

### 8.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −38.

 0: −1 − 4⋅i4_0 2: 1 − 4⋅i4_0 2_var_snapshot: −4⋅i4_0 2*: 2 − 4⋅i4_0
Hints:
 16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ] 18 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ] 1 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ]

### 8.2.2 Transition Removal

We remove transitions 16, 18 using the following ranking functions, which are bounded by −1.

 0: −2 2: 0 2_var_snapshot: −1 2*: 1
Hints:
 16 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 18 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 8.2.3 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −1.

 0: −1 2: 0 2_var_snapshot: 0 2*: 0
Hints:
 2 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 8.2.4 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

### 8.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 15.

### 8.2.4.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

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