# 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 ∧ − x2 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 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 1 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − x7 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ −99 + x7 − 100⋅x8 ≤ 0 ∧ − x7 + 100⋅x8 ≤ 0 ∧ − arg1P + x7 − 100⋅x8 ≤ 0 ∧ arg1P − x7 + 100⋅x8 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − x2 + x2 ≤ 0 ∧ x2 − x2 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 2 2 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ −1 − arg1P + arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x2 + x2 ≤ 0 ∧ x2 − x2 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 3 3 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x2 + x2 ≤ 0 ∧ x2 − x2 ≤ 0

## Proof

The following invariants are asserted.

 0: TRUE 1: 1 − arg1 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ − x2 ≤ 0 2: 1 − arg2 ≤ 0 ∧ − x2 ≤ 0 ∧ − x7 ≤ 0 3: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) 1 − arg1 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ − x2 ≤ 0 2 (2) 1 − arg2 ≤ 0 ∧ − x2 ≤ 0 ∧ − x7 ≤ 0 3 (3) TRUE
• initial node: 3
• cover edges:
• transition edges:  0 0 1 1 1 2 2 2 2 3 3 0

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 2 4 2: − x8 + x8 ≤ 0 ∧ x8 − x8 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x2 + x2 ≤ 0 ∧ x2 − x2 ≤ 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, 1, 3 using the following ranking functions, which are bounded by −13.

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

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

2* 7 2: x8 + x8 ≤ 0x8x8 ≤ 0x7 + x7 ≤ 0x7x7 ≤ 0x2 + x2 ≤ 0x2x2 ≤ 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 5 2_var_snapshot: x8 + x8 ≤ 0x8x8 ≤ 0x7 + x7 ≤ 0x7x7 ≤ 0x2 + x2 ≤ 0x2x2 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 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 { 2, 2_var_snapshot, 2* }.

### 6.1.1 Transition Removal

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

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

### 6.1.2 Transition Removal

We remove transitions 5, 7 using the following ranking functions, which are bounded by −1.

 2: 0 2_var_snapshot: −1 2*: arg2

### 6.1.3 Splitting Cut-Point Transitions

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

### 6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 4.

### 6.1.3.1.1 Splitting Cut-Point Transitions

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

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