# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 3
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0 ∧ −1 − arg1P + arg2 ≤ 0 ∧ 1 + arg1P − arg2 ≤ 0 ∧ − arg3P ≤ 0 ∧ arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 1 1 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ −1 − arg1P + arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ arg1 − arg2P ≤ 0 ∧ − arg1 + arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 1 2 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ −1 − arg1P + arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ arg1 − arg2P ≤ 0 ∧ − arg1 + arg2P ≤ 0 ∧ 1 − arg3P ≤ 0 ∧ −1 + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 1 3 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ arg2 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 2 4 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 3 5 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 6 1: − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 2 13 2: − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
and for every transition t, a duplicate t is considered.

### 2 Transition Removal

We remove transitions 0, 3, 5 using the following ranking functions, which are bounded by −15.

 3: 0 0: 0 1: 0 2: 0 3: −5 0: −6 1: −7 1_var_snapshot: −7 1*: −7 2: −10 2_var_snapshot: −10 2*: −10

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

1* 9 1: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

1 7 1_var_snapshot: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

2* 16 2: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

2 14 2_var_snapshot: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

### 7 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

### 7.1 SCC Subproblem 1/2

Here we consider the SCC { 2, 2_var_snapshot, 2* }.

### 7.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 2.

 2: 1 + 3⋅arg1 2_var_snapshot: 3⋅arg1 2*: 2 + 3⋅arg1

### 7.1.2 Transition Removal

We remove transition 14 using the following ranking functions, which are bounded by −1.

 2: 0 2_var_snapshot: −1 2*: 1

### 7.1.3 Transition Removal

We remove transition 16 using the following ranking functions, which are bounded by −1.

 2: −1 2_var_snapshot: 0 2*: 0

### 7.1.4 Splitting Cut-Point Transitions

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

### 7.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 13.

### 7.1.4.1.1 Splitting Cut-Point Transitions

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

### 7.2 SCC Subproblem 2/2

Here we consider the SCC { 1, 1_var_snapshot, 1* }.

### 7.2.1 Splitting Cut-Point Transitions

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

### 7.2.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 6.

The new variable __snapshot_1_arg3P is introduced. The transition formulas are extended as follows:

 7: __snapshot_1_arg3P ≤ arg3P ∧ arg3P ≤ __snapshot_1_arg3P 9: __snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P 1: __snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P 2: __snapshot_1_arg3P ≤ __snapshot_1_arg3P ∧ __snapshot_1_arg3P ≤ __snapshot_1_arg3P

The new variable __snapshot_1_arg3 is introduced. The transition formulas are extended as follows:

 7: __snapshot_1_arg3 ≤ arg3 ∧ arg3 ≤ __snapshot_1_arg3 9: __snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3 1: __snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3 2: __snapshot_1_arg3 ≤ __snapshot_1_arg3 ∧ __snapshot_1_arg3 ≤ __snapshot_1_arg3

The new variable __snapshot_1_arg2P is introduced. The transition formulas are extended as follows:

 7: __snapshot_1_arg2P ≤ arg2P ∧ arg2P ≤ __snapshot_1_arg2P 9: __snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P 1: __snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P 2: __snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P

The new variable __snapshot_1_arg2 is introduced. The transition formulas are extended as follows:

 7: __snapshot_1_arg2 ≤ arg2 ∧ arg2 ≤ __snapshot_1_arg2 9: __snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2 1: __snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2 2: __snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2

The new variable __snapshot_1_arg1P is introduced. The transition formulas are extended as follows:

 7: __snapshot_1_arg1P ≤ arg1P ∧ arg1P ≤ __snapshot_1_arg1P 9: __snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P 1: __snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P 2: __snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P

The new variable __snapshot_1_arg1 is introduced. The transition formulas are extended as follows:

 7: __snapshot_1_arg1 ≤ arg1 ∧ arg1 ≤ __snapshot_1_arg1 9: __snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1 1: __snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1 2: __snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1

The following invariants are asserted.

 0: arg2 − arg2P ≤ 0 1: −1 − arg1 + arg2 ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − arg2P ≤ 0 2: TRUE 3: TRUE 1: −1 − arg1 + arg2 ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − arg2P ≤ 0 ∨ −1 − arg1 + arg2 ≤ 0 ∧ 1 − __snapshot_1_arg2P + arg2P ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − __snapshot_1_arg2P ≤ 0 ∧ − arg2P ≤ 0 1_var_snapshot: 1 − __snapshot_1_arg2P + arg1 ≤ 0 ∧ −1 − arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg2P ≤ 0 1*: −1 − arg1 + arg2 ≤ 0 ∧ 1 − __snapshot_1_arg2P + arg2P ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − __snapshot_1_arg2P ≤ 0 ∧ − arg2P ≤ 0

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (3) TRUE 1 (0) arg2 − arg2P ≤ 0 2 (1) −1 − arg1 + arg2 ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − arg2P ≤ 0 3 (1) −1 − arg1 + arg2 ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − arg2P ≤ 0 4 (1) −1 − arg1 + arg2 ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − arg2P ≤ 0 5 (2) TRUE 6 (1) −1 − arg1 + arg2 ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − arg2P ≤ 0 7 (1_var_snapshot) 1 − __snapshot_1_arg2P + arg1 ≤ 0 ∧ −1 − arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg2P ≤ 0 12 (1*) −1 − arg1 + arg2 ≤ 0 ∧ 1 − __snapshot_1_arg2P + arg2P ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − __snapshot_1_arg2P ≤ 0 ∧ − arg2P ≤ 0 13 (1*) −1 − arg1 + arg2 ≤ 0 ∧ 1 − __snapshot_1_arg2P + arg2P ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − __snapshot_1_arg2P ≤ 0 ∧ − arg2P ≤ 0 14 (1) −1 − arg1 + arg2 ≤ 0 ∧ 1 − __snapshot_1_arg2P + arg2P ≤ 0 ∧ 1 + arg1 − arg2P ≤ 0 ∧ − __snapshot_1_arg2P ≤ 0 ∧ − arg2P ≤ 0 15 (1_var_snapshot) 1 − __snapshot_1_arg2P + arg1 ≤ 0 ∧ −1 − arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg2P ≤ 0 20 (2) TRUE
• initial node: 0
• cover edges:  3 → 2 4 → 2 12 → 13 15 → 7 20 → 5
• transition edges:  0 5 1 1 0 2 2 1 3 2 2 4 2 3 5 2 6 6 5 4 20 6 7 7 7 1 12 7 2 13 13 9 14 14 7 15

### 7.2.1.1.8 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by −2.

 1: arg2P 1_var_snapshot: __snapshot_1_arg2P 1*: __snapshot_1_arg2P

### 7.2.1.1.9 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −5.

 1: −1 1_var_snapshot: −2 1*: −3

### 7.2.1.1.10 Splitting Cut-Point Transitions

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

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