# 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 ∧ − x_0 ≤ 0 ∧ Result_0 − Result_post ≤ 0 ∧ − Result_0 + Result_post ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 0 1 2: 1 + x_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0 2 2 3: 2 + y_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0 2 3 3: − y_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0 3 4 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − x_0 + x_post ≤ 0 ∧ 1 + x_0 − x_post ≤ 0 ∧ −1 − y_0 + y_post ≤ 0 ∧ 1 + y_0 − y_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0 4 5 0: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0 0 6 5: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_0 ≤ 0 ∧ 1 + y_0 ≤ 0 ∧ −1 − y_0 ≤ 0 ∧ −1 − y_0 + y_post ≤ 0 ∧ 1 + y_0 − y_post ≤ 0 ∧ 99 − x_0 + x_post ≤ 0 ∧ −99 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0 5 7 0: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0 6 8 0: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0 7 9 6: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 0 10 0: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 2 Transition Removal

We remove transitions 0, 8, 9 using the following ranking functions, which are bounded by −13.

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

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

0* 13 0: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0Result_post + Result_post ≤ 0Result_postResult_post ≤ 0Result_0 + Result_0 ≤ 0Result_0Result_0 ≤ 0

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

0 11 0_var_snapshot: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0Result_post + Result_post ≤ 0Result_postResult_post ≤ 0Result_0 + Result_0 ≤ 0Result_0Result_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, 2, 3, 4, 5, 0_var_snapshot, 0* }.

### 5.1.1 Transition Removal

We remove transitions 2, 6 using the following ranking functions, which are bounded by 2.

 0: −2 − 6⋅y_0 2: −4 − 6⋅y_0 3: −5 − 6⋅y_0 4: −6⋅y_0 5: −6⋅y_0 0_var_snapshot: −3 − 6⋅y_0 0*: −1 − 6⋅y_0
Hints:
 11 lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ] 13 lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexWeak[ [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ] 2 lexStrict[ [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] , [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 3 lexWeak[ [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ] 4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0] ] 5 lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ] 6 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 7 lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 5.1.2 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 2.

 0: −2 − 5⋅x_0 2: −3 − 5⋅x_0 3: −4 − 5⋅x_0 4: −5⋅x_0 5: −5⋅x_0 0_var_snapshot: −2 − 5⋅x_0 0*: −1 − 5⋅x_0
Hints:
 11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ] 13 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ] 1 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] , [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ] 4 lexWeak[ [0, 0, 0, 0, 0, 5, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0] ] 5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ] 7 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]

### 5.1.3 Transition Removal

We remove transitions 11, 13, 3, 4, 5, 7 using the following ranking functions, which are bounded by −3.

 0: −2 2: 2 3: 1 4: 0 5: 0 0_var_snapshot: −3 0*: −1
Hints:
 11 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] ] 13 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] ] 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] ] 4 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] ] 5 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] ] 7 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] ]

### 5.1.4 Splitting Cut-Point Transitions

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

### 5.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 10.

### 5.1.4.1.1 Splitting Cut-Point Transitions

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

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