# 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 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ −1 + arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 1 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ −99 + arg1 ≤ 0 ∧ − arg2P ≤ 0 ∧ arg2P ≤ 0 ∧ 100 − arg1 − arg3P ≤ 0 ∧ −100 + arg1 + arg3P ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 2 2 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2 + arg3 ≤ 0 ∧ 1 − arg1P + arg1 ≤ 0 ∧ −1 + arg1P − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 2 3 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −99 + arg2 ≤ 0 ∧ 1 + arg2 − arg3 ≤ 0 ∧ − arg2 ≤ 0 ∧ −98 + arg2 ≤ 0 ∧ − x7 + x8 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg2P + arg2 ≤ 0 ∧ −1 + arg2P − arg2 ≤ 0 ∧ 100 − arg1 − arg3P ≤ 0 ∧ −100 + arg1 + arg3P ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 2 4 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −99 + arg2 ≤ 0 ∧ 1 + arg2 − arg3 ≤ 0 ∧ − arg2 ≤ 0 ∧ −98 + arg2 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 + x12 − x13 ≤ 0 ∧ 1 − arg2P + arg2 ≤ 0 ∧ −1 + arg2P − arg2 ≤ 0 ∧ 100 − arg1 − arg3P ≤ 0 ∧ −100 + arg1 + arg3P ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 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 ∧ − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0

## Proof

The following invariants are asserted.

 0: TRUE 1: TRUE 2: −99 + arg3P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ −99 + arg3 ≤ 0 3: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) TRUE 2 (2) −99 + arg3P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ −99 + arg3 ≤ 0 3 (3) TRUE
• initial node: 3
• cover edges:
• transition edges:  0 0 1 1 1 2 2 2 1 2 3 2 2 4 2 3 5 0

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 6 1: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − 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: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − 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.

### 3 Transition Removal

We remove transitions 0, 5 using the following ranking functions, which are bounded by −13.

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

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

1* 9 1: x8 + x8 ≤ 0x8x8 ≤ 0x7 + x7 ≤ 0x7x7 ≤ 0x13 + x13 ≤ 0x13x13 ≤ 0x12 + x12 ≤ 0x12x12 ≤ 0arg3P + 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: x8 + x8 ≤ 0x8x8 ≤ 0x7 + x7 ≤ 0x7x7 ≤ 0x13 + x13 ≤ 0x13x13 ≤ 0x12 + x12 ≤ 0x12x12 ≤ 0arg3P + 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: x8 + x8 ≤ 0x8x8 ≤ 0x7 + x7 ≤ 0x7x7 ≤ 0x13 + x13 ≤ 0x13x13 ≤ 0x12 + x12 ≤ 0x12x12 ≤ 0arg3P + 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: x8 + x8 ≤ 0x8x8 ≤ 0x7 + x7 ≤ 0x7x7 ≤ 0x13 + x13 ≤ 0x13x13 ≤ 0x12 + x12 ≤ 0x12x12 ≤ 0arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

### 8 SCC Decomposition

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

### 8.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 2, 1_var_snapshot, 1*, 2_var_snapshot, 2* }.

### 8.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −396.

 1: 2 − 4⋅arg1 2: −4⋅arg1 1_var_snapshot: 1 − 4⋅arg1 1*: 3 − 4⋅arg1 2_var_snapshot: −4⋅arg1 2*: −4⋅arg1

### 8.1.2 Transition Removal

We remove transitions 9, 2 using the following ranking functions, which are bounded by −3.

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

### 8.1.3 Transition Removal

We remove transitions 7, 3, 4 using the following ranking functions, which are bounded by −298.

 1: 1 2: 1 − 3⋅arg2 1_var_snapshot: 0 1*: 0 2_var_snapshot: −3⋅arg2 2*: 2 − 3⋅arg2

### 8.1.4 Transition Removal

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

 1: 0 2: 0 1_var_snapshot: 0 1*: 0 2_var_snapshot: − arg1 2*: 1

### 8.1.5 Transition Removal

We remove transition 16 using the following ranking functions, which are bounded by 0.

 1: 0 2: 0 1_var_snapshot: 0 1*: 0 2_var_snapshot: 0 2*: arg1

### 8.1.6 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

### 8.1.6.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 6.

### 8.1.6.1.1 Splitting Cut-Point Transitions

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

### 8.1.6.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 13.

### 8.1.6.2.1 Splitting Cut-Point Transitions

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

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