# 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 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 0 1 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 2 2 0: 10 − i_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 2 3 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + i_0 ≤ 0 ∧ −1 − i_0 + i_post ≤ 0 ∧ 1 + i_0 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 3 4 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 1 5 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 5 6 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 6 7 5: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

## Proof

The following invariants are asserted.

 0: 10 − i_0 ≤ 0 1: 10 − i_0 ≤ 0 2: TRUE 3: TRUE 4: 10 − i_0 ≤ 0 5: TRUE 6: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) 10 − i_0 ≤ 0 1 (1) 10 − i_0 ≤ 0 2 (2) TRUE 3 (3) TRUE 4 (4) 10 − i_0 ≤ 0 5 (5) TRUE 6 (6) TRUE
• initial node: 6
• cover edges:
• transition edges:  0 0 1 0 1 1 1 5 4 2 2 0 2 3 3 3 4 2 5 6 3 6 7 5

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 3 8 3: − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 1, 2, 5, 6, 7 using the following ranking functions, which are bounded by −17.

 6: 0 5: 0 2: 0 3: 0 0: 0 1: 0 4: 0 6: −7 5: −8 2: −9 3: −9 3_var_snapshot: −9 3*: −9 0: −10 1: −11 4: −12
Hints:
 9 lexWeak[ [0, 0, 0, 0] ] 3 lexWeak[ [0, 0, 0, 0, 0, 0, 0] ] 4 lexWeak[ [0, 0, 0, 0, 0, 0] ] 0 lexStrict[ [0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0] ] 1 lexStrict[ [0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0] ] 2 lexStrict[ [0, 0, 0, 0, 0] , [0, 0, 0, 0, 0] ] 5 lexStrict[ [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] ] 7 lexStrict[ [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.

3* 11 3: i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0

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

3 9 3_var_snapshot: i_post + i_post ≤ 0i_posti_post ≤ 0i_0 + i_0 ≤ 0i_0i_0 ≤ 0

### 6 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

### 6.1 SCC Subproblem 1/1

Here we consider the SCC { 2, 3, 3_var_snapshot, 3* }.

### 6.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −38.

 2: −1 − 4⋅i_0 3: 1 − 4⋅i_0 3_var_snapshot: −4⋅i_0 3*: 2 − 4⋅i_0
Hints:
 9 lexWeak[ [0, 0, 0, 4] ] 11 lexWeak[ [0, 0, 0, 4] ] 3 lexStrict[ [0, 0, 0, 0, 4, 0, 4] , [0, 0, 4, 0, 0, 0, 0] ] 4 lexWeak[ [0, 0, 0, 0, 0, 4] ]

### 6.1.2 Transition Removal

We remove transitions 11, 4 using the following ranking functions, which are bounded by −2.

 2: −2 3: 0 3_var_snapshot: −1 3*: 1
Hints:
 9 lexWeak[ [0, 0, 0, 0] ] 11 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ] 4 lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]

### 6.1.3 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

 2: 0 3: 1 3_var_snapshot: 0 3*: 0
Hints:
 9 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]

### 6.1.4 Splitting Cut-Point Transitions

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

### 6.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

### 6.1.4.1.1 Splitting Cut-Point Transitions

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

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