# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 7
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − i1_0 + i1_post ≤ 0 ∧ 1 + i1_0 − i1_post ≤ 0 ∧ i1_0 − i1_post ≤ 0 ∧ − i1_0 + i1_post ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0 2 1 0: − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0 3 2 2: − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0 3 3 0: − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0 3 4 2: − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0 4 5 5: ___const_42_0 − i1_0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0 4 6 3: 1 − ___const_42_0 + i1_0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0 1 7 4: − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0 6 8 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ i1_post ≤ 0 ∧ − i1_post ≤ 0 ∧ i1_0 − i1_post ≤ 0 ∧ − i1_0 + i1_post ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0 7 9 6: − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 10 1: − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 ∧ − ___const_42_0 + ___const_42_0 ≤ 0 ∧ ___const_42_0 − ___const_42_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 2 Transition Removal

We remove transitions 5, 8, 9 using the following ranking functions, which are bounded by −13.

 7: 0 6: 0 0: 0 1: 0 2: 0 3: 0 4: 0 5: 0 7: −5 6: −6 0: −7 1: −7 2: −7 3: −7 4: −7 1_var_snapshot: −7 1*: −7 5: −8

### 3 Location Addition

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

1* 13 1: i1_post + i1_post ≤ 0i1_posti1_post ≤ 0i1_0 + i1_0 ≤ 0i1_0i1_0 ≤ 0___const_42_0 + ___const_42_0 ≤ 0___const_42_0___const_42_0 ≤ 0

### 4 Location Addition

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

1 11 1_var_snapshot: i1_post + i1_post ≤ 0i1_posti1_post ≤ 0i1_0 + i1_0 ≤ 0i1_0i1_0 ≤ 0___const_42_0 + ___const_42_0 ≤ 0___const_42_0___const_42_0 ≤ 0

### 5 SCC Decomposition

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

### 5.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 1, 2, 3, 4, 1_var_snapshot, 1* }.

### 5.1.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 5.

 0: −4 + 7⋅___const_42_0 − 7⋅i1_0 1: 1 + 7⋅___const_42_0 − 7⋅i1_0 2: −3 + 7⋅___const_42_0 − 7⋅i1_0 3: −2 + 7⋅___const_42_0 − 7⋅i1_0 4: −1 + 7⋅___const_42_0 − 7⋅i1_0 1_var_snapshot: 7⋅___const_42_0 − 7⋅i1_0 1*: 2 + 7⋅___const_42_0 − 7⋅i1_0

### 5.1.2 Transition Removal

We remove transitions 11, 13, 0, 1, 2, 3, 4, 7 using the following ranking functions, which are bounded by −6.

 0: −2 1: −4 2: −1 3: 0 4: −6 1_var_snapshot: −5 1*: −3

### 5.1.3 Splitting Cut-Point Transitions

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

### 5.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 10.

### 5.1.3.1.1 Splitting Cut-Point Transitions

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

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