# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 3
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P ≤ 0 ∧ arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 1 1 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −99 + arg1 ≤ 0 ∧ 1 − arg1P + arg1 ≤ 0 ∧ −1 + arg1P − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 1 2 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 100 − arg1 ≤ 0 ∧ 5 − arg1P ≤ 0 ∧ −5 + arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 2 3 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −20 + arg1 ≤ 0 ∧ 3 − arg1P + arg1 ≤ 0 ∧ −3 + arg1P − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 3 4 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 5 1: − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 2 12 2: − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
and for every transition t, a duplicate t is considered.

### 2 Transition Removal

We remove transitions 0, 2, 4 using the following ranking functions, which are bounded by −15.

 3: 0 0: 0 1: 0 2: 0 3: −5 0: −6 1: −7 1_var_snapshot: −7 1*: −7 2: −10 2_var_snapshot: −10 2*: −10
Hints:
 6 lexWeak[ [0, 0, 0, 0] ] 13 lexWeak[ [0, 0, 0, 0] ] 1 lexWeak[ [0, 0, 0, 0, 0, 0, 0] ] 3 lexWeak[ [0, 0, 0, 0, 0, 0, 0] ] 0 lexStrict[ [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] ] 4 lexStrict[ [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* 8 1: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

1 6 1_var_snapshot: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

2* 15 2: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

2 13 2_var_snapshot: arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

### 7 SCC Decomposition

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

### 7.1 SCC Subproblem 1/2

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

### 7.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −21.

 2: 1 − arg1 2_var_snapshot: − arg1 2*: 2 − arg1
Hints:
 13 lexWeak[ [0, 0, 0, 1] ] 15 lexWeak[ [0, 0, 0, 1] ] 3 lexStrict[ [0, 0, 0, 1, 0, 0, 1] , [0, 0, 1, 0, 0, 0, 0] ]

### 7.1.2 Transition Removal

We remove transitions 13, 15 using the following ranking functions, which are bounded by −2.

 2: −1 2_var_snapshot: −2 2*: 0
Hints:
 13 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ] 15 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]

### 7.1.3 Splitting Cut-Point Transitions

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

### 7.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 12.

### 7.1.3.1.1 Splitting Cut-Point Transitions

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

### 7.2 SCC Subproblem 2/2

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

### 7.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −298.

 1: 1 − 3⋅arg1 1_var_snapshot: −3⋅arg1 1*: 2 − 3⋅arg1
Hints:
 6 lexWeak[ [0, 0, 0, 3] ] 8 lexWeak[ [0, 0, 0, 3] ] 1 lexStrict[ [0, 0, 0, 3, 0, 0, 3] , [0, 0, 3, 0, 0, 0, 0] ]

### 7.2.2 Transition Removal

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

 1: 0 1_var_snapshot: −1 1*: 1
Hints:
 6 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ] 8 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]

### 7.2.3 Splitting Cut-Point Transitions

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

### 7.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

### 7.2.3.1.1 Splitting Cut-Point Transitions

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

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