# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 6
• Transitions: (pre-variables and post-variables)  2 1 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 1 2 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2P ≤ 0 ∧ 1 − x7 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 3 3 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ −1 + arg2 ≤ 0 ∧ 2 + arg1P − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 3 4 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ 2 − arg2 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P ≤ 0 ∧ −1 − arg2P + arg2 ≤ 0 ∧ 1 + arg2P − arg2 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 4 5 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ −1 − arg1P + arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 2 6 5: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg2P ≤ 0 ∧ −1 + arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 5 7 5: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ −1 − arg1P + arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ 1 − arg2P + arg2 ≤ 0 ∧ −1 + arg2P − arg2 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 6 8 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0

## Proof

The following invariants are asserted.

 1: − arg1P ≤ 0 ∧ − arg1 ≤ 0 2: TRUE 3: 1 − x7 ≤ 0 4: 1 − x7 ≤ 0 5: TRUE 6: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  1 (1) − arg1P ≤ 0 ∧ − arg1 ≤ 0 2 (2) TRUE 3 (3) 1 − x7 ≤ 0 4 (4) 1 − x7 ≤ 0 5 (5) TRUE 6 (6) TRUE
• initial node: 6
• cover edges:
• transition edges:  1 2 3 2 1 1 2 6 5 3 3 4 3 4 3 4 5 4 5 7 5 6 8 2

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 3 9 3: − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 4 16 4: − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 5 23 5: − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 1, 2, 3, 6, 8 using the following ranking functions, which are bounded by −21.

 6: 0 2: 0 1: 0 3: 0 4: 0 5: 0 6: −7 2: −8 1: −9 3: −10 3_var_snapshot: −10 3*: −10 4: −13 4_var_snapshot: −13 4*: −13 5: −16 5_var_snapshot: −16 5*: −16
Hints:
 10 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 17 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 24 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 4 lexWeak[ [0, 0, 0, 0, 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] ] 7 lexWeak[ [0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 3 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, 0, 0] ] 8 lexStrict[ [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.

3* 12 3: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

3 10 3_var_snapshot: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

4* 19 4: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

4 17 4_var_snapshot: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

5* 26 5: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

5 24 5_var_snapshot: x7 + x7 ≤ 0x7x7 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

### 10 SCC Decomposition

We consider subproblems for each of the 3 SCC(s) of the program graph.

### 10.1 SCC Subproblem 1/3

Here we consider the SCC { 4, 4_var_snapshot, 4* }.

### 10.1.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 2.

 4: 1 + 3⋅arg1 4_var_snapshot: 3⋅arg1 4*: 2 + 3⋅arg1
Hints:
 17 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0] ] 19 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0] ] 5 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 10.1.2 Transition Removal

We remove transitions 17, 19 using the following ranking functions, which are bounded by −1.

 4: 0 4_var_snapshot: −1 4*: 1
Hints:
 17 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 19 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 10.1.3 Splitting Cut-Point Transitions

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

### 10.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 16.

### 10.1.3.1.1 Splitting Cut-Point Transitions

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

### 10.2 SCC Subproblem 2/3

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

### 10.2.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 3.

 3: 1 + 2⋅arg1 + arg2 3_var_snapshot: 2⋅arg1 + arg2 3*: 2 + 2⋅arg1 + arg2
Hints:
 10 lexWeak[ [0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0] ] 12 lexWeak[ [0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0] ] 4 lexStrict[ [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 10.2.2 Transition Removal

We remove transitions 10, 12 using the following ranking functions, which are bounded by −1.

 3: 0 3_var_snapshot: −1 3*: x7
Hints:
 10 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 12 lexStrict[ [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 10.2.3 Splitting Cut-Point Transitions

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

### 10.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 9.

### 10.2.3.1.1 Splitting Cut-Point Transitions

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

### 10.3 SCC Subproblem 3/3

Here we consider the SCC { 5, 5_var_snapshot, 5* }.

### 10.3.1 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by 2.

 5: 1 + 3⋅arg1 5_var_snapshot: 3⋅arg1 5*: 2 + 3⋅arg1
Hints:
 24 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0] ] 26 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0] ] 7 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 10.3.2 Transition Removal

We remove transitions 24, 26 using the following ranking functions, which are bounded by −2.

 5: −1 5_var_snapshot: −2 5*: 0
Hints:
 24 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 26 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 10.3.3 Splitting Cut-Point Transitions

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

### 10.3.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 23.

### 10.3.3.1.1 Splitting Cut-Point Transitions

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

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