# 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_5_0 + y_6_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_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 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_0 ≤ 0 0 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ tmp_7_post ≤ 0 ∧ − tmp_7_post ≤ 0 ∧ 1 − y_6_0 + y_6_post ≤ 0 ∧ −1 + y_6_0 − y_6_post ≤ 0 ∧ tmp_7_0 − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_post ≤ 0 ∧ y_6_0 − y_6_post ≤ 0 ∧ − y_6_0 + y_6_post ≤ 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 0: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_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 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_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 0 3 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ tmp_7_0 − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_post ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_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 ∧ − 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 4 4: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_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 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_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 4 5 5: 0 ≤ 0 ∧ 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_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_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 6 0: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_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 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_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 0: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_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 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_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 6: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_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 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_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

The following invariants are asserted.

 0: TRUE 1: TRUE 2: tmp_7_post ≤ 0 ∧ − tmp_7_post ≤ 0 ∧ tmp_7_0 ≤ 0 ∧ − tmp_7_0 ≤ 0 3: TRUE 4: TRUE 5: TRUE 6: TRUE 7: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) TRUE 2 (2) tmp_7_post ≤ 0 ∧ − tmp_7_post ≤ 0 ∧ tmp_7_0 ≤ 0 ∧ − tmp_7_0 ≤ 0 3 (3) TRUE 4 (4) TRUE 5 (5) TRUE 6 (6) TRUE 7 (7) TRUE
• initial node: 7
• cover edges:
• transition edges:  0 0 1 0 1 2 0 3 3 2 2 0 3 4 4 4 5 5 5 6 0 6 7 0 7 8 6

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 0 9 0: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_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 ∧ − tmp_7_post + tmp_7_post ≤ 0 ∧ tmp_7_post − tmp_7_post ≤ 0 ∧ − tmp_7_0 + tmp_7_0 ≤ 0 ∧ tmp_7_0 − tmp_7_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.

### 3 Transition Removal

We remove transitions 0, 7, 8 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

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

0* 12 0: y_6_post + y_6_post ≤ 0y_6_posty_6_post ≤ 0y_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 ≤ 0tmp_7_post + tmp_7_post ≤ 0tmp_7_posttmp_7_post ≤ 0tmp_7_0 + tmp_7_0 ≤ 0tmp_7_0tmp_7_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.

0 10 0_var_snapshot: y_6_post + y_6_post ≤ 0y_6_posty_6_post ≤ 0y_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 ≤ 0tmp_7_post + tmp_7_post ≤ 0tmp_7_posttmp_7_post ≤ 0tmp_7_0 + tmp_7_0 ≤ 0tmp_7_0tmp_7_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 0

### 6 SCC Decomposition

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

### 6.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 2, 3, 4, 5, 0_var_snapshot, 0* }.

### 6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 0.

 0: −2 − 3⋅x_5_0 + 3⋅y_6_0 2: −1 − 3⋅x_5_0 + 3⋅y_6_0 3: −2 − 3⋅x_5_0 + 3⋅y_6_0 4: −3 − 3⋅x_5_0 + 3⋅y_6_0 5: −3⋅x_5_0 + 3⋅y_6_0 0_var_snapshot: −2 − 3⋅x_5_0 + 3⋅y_6_0 0*: −1 − 3⋅x_5_0 + 3⋅y_6_0

### 6.1.2 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 2.

 0: −2 − 6⋅x_5_0 + 6⋅y_6_0 2: −6⋅x_5_0 + 6⋅y_6_0 3: −4 − 6⋅x_5_0 + 6⋅y_6_0 4: −5 − 6⋅x_5_0 + 6⋅y_6_0 5: −6⋅x_5_0 + 6⋅y_6_0 0_var_snapshot: −3 − 6⋅x_5_0 + 6⋅y_6_0 0*: −1 − 6⋅x_5_0 + 6⋅y_6_0

### 6.1.3 Transition Removal

We remove transitions 10, 12, 2, 5, 6 using the following ranking functions, which are bounded by −2.

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

### 6.1.4 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −1.

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

### 6.1.5 Splitting Cut-Point Transitions

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

### 6.1.5.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 9.

### 6.1.5.1.1 Splitting Cut-Point Transitions

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

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