# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 4
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − x_5_0 + x_5_post ≤ 0 ∧ 1 + x_5_0 − x_5_post ≤ 0 ∧ −1 − y_6_0 + y_6_post ≤ 0 ∧ 1 + y_6_0 − y_6_post ≤ 0 ∧ 1 − z_7_0 + z_7_post ≤ 0 ∧ −1 + z_7_0 − z_7_post ≤ 0 ∧ x_5_0 − x_5_post ≤ 0 ∧ − x_5_0 + x_5_post ≤ 0 ∧ y_6_0 − y_6_post ≤ 0 ∧ − y_6_0 + y_6_post ≤ 0 ∧ z_7_0 − z_7_post ≤ 0 ∧ − z_7_0 + z_7_post ≤ 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 0: − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 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 2 2 3: 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 ∧ − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 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 2 3 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ − y_6_0 + z_7_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 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 2 4 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ 1 + y_6_0 − z_7_0 ≤ 0 ∧ 1 + x_5_0 − z_7_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 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 2 5 0: 1 + x_5_0 − y_6_0 ≤ 0 ∧ 1 + y_6_0 − z_7_0 ≤ 0 ∧ − x_5_0 + z_7_0 ≤ 0 ∧ − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 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 4 6 2: − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 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

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 0 7 0: − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 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
and for every transition t, a duplicate t is considered.

### 2 Transition Removal

We remove transitions 2, 3, 4, 5, 6 using the following ranking functions, which are bounded by −13.

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

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

0* 10 0: z_7_post + z_7_post ≤ 0z_7_postz_7_post ≤ 0z_7_0 + z_7_0 ≤ 0z_7_0z_7_0 ≤ 0y_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 ≤ 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 8 0_var_snapshot: z_7_post + z_7_post ≤ 0z_7_postz_7_post ≤ 0z_7_0 + z_7_0 ≤ 0z_7_0z_7_0 ≤ 0y_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 ≤ 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 { 0, 1, 0_var_snapshot, 0* }.

### 5.1.1 Splitting Cut-Point Transitions

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

### 5.1.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

The following invariants are asserted.

 0: 1 ≤ 0 1: 1 ≤ 0 2: TRUE 3: TRUE 4: TRUE 0: 1 ≤ 0 1: 1 ≤ 0 0_var_snapshot: 1 ≤ 0 0*: 1 ≤ 0

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (4) TRUE 1 (2) TRUE 2 (3) TRUE 3 (3) TRUE 4 (3) TRUE 5 (0) 1 ≤ 0
• initial node: 0
• cover edges:  2 → 4 3 → 4
• transition edges:  0 6 1 1 2 2 1 3 3 1 4 4 1 5 5

### 5.1.1.1.2 Transition Removal

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

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

### 5.1.1.1.3 Splitting Cut-Point Transitions

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

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