# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 7
• Transitions: (pre-variables and post-variables)  0 0 1: − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0 1 1 2: − x_5_0 + y_6_0 ≤ 0 ∧ − x_5_0 + y_6_0 ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0 2 2 3: − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0 3 3 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 1 4 5: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_5_0 + y_6_0 ≤ 0 ∧ − x_5_0 + y_6_0 ≤ 0 ∧ x_5_0 − y_6_0 ≤ 0 ∧ − x_5_0 + y_6_0 ≤ 0 ∧ −1 − x_5_0 + x_5_post ≤ 0 ∧ 1 + x_5_0 − x_5_post ≤ 0 ∧ x_5_0 − x_5_post ≤ 0 ∧ − x_5_0 + x_5_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0 5 5 1: − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0 1 6 6: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ −1 − x_5_0 + x_5_post ≤ 0 ∧ 1 + x_5_0 − x_5_post ≤ 0 ∧ x_5_0 − x_5_post ≤ 0 ∧ − x_5_0 + x_5_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0 6 7 1: − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0 7 8 0: − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 9 1: − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 2 Transition Removal

We remove transitions 0, 1, 2, 3, 8 using the following ranking functions, which are bounded by −17.

 7: 0 0: 0 1: 0 5: 0 6: 0 2: 0 3: 0 4: 0 7: −7 0: −8 1: −9 5: −9 6: −9 1_var_snapshot: −9 1*: −9 2: −13 3: −14 4: −15
Hints:
 10 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] ] 5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 6 lexWeak[ [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 lexStrict[ [0, 0, 0, 0, 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] ] 3 lexStrict[ [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.

1* 12 1: y_6_0 + y_6_0 ≤ 0y_6_0y_6_0 ≤ 0x_5_post + x_5_post ≤ 0x_5_postx_5_post ≤ 0x_5_0 + x_5_0 ≤ 0x_5_0x_5_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 0

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

1 10 1_var_snapshot: y_6_0 + y_6_0 ≤ 0y_6_0y_6_0 ≤ 0x_5_post + x_5_post ≤ 0x_5_postx_5_post ≤ 0x_5_0 + x_5_0 ≤ 0x_5_0x_5_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_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 { 1, 5, 6, 1_var_snapshot, 1* }.

### 5.1.1 Transition Removal

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

 1: −2 − 4⋅x_5_0 + 4⋅y_6_0 5: −4⋅x_5_0 + 4⋅y_6_0 6: −4⋅x_5_0 + 4⋅y_6_0 1_var_snapshot: −3 − 4⋅x_5_0 + 4⋅y_6_0 1*: −1 − 4⋅x_5_0 + 4⋅y_6_0
Hints:
 10 lexWeak[ [4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ] 12 lexWeak[ [4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ] 4 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 5 lexWeak[ [4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ] 6 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0] , [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 7 lexWeak[ [4, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]

### 5.1.2 Transition Removal

We remove transitions 10, 12, 5, 7 using the following ranking functions, which are bounded by −3.

 1: −2 5: 0 6: 0 1_var_snapshot: −3 1*: −1
Hints:
 10 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 12 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [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, 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] ]

### 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 9.

### 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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