# 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 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ 1 − b_7_0 ≤ 0 ∧ b_7_post ≤ 0 ∧ − b_7_post ≤ 0 ∧ 1 − y_6_0 + y_6_post ≤ 0 ∧ −1 + y_6_0 − y_6_post ≤ 0 ∧ b_7_0 − b_7_post ≤ 0 ∧ − b_7_0 + b_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 0 1 2: 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 ∧ − b_7_post + b_7_post ≤ 0 ∧ b_7_post − b_7_post ≤ 0 ∧ − b_7_0 + b_7_0 ≤ 0 ∧ b_7_0 − b_7_0 ≤ 0 3 2 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ b_7_post ≤ 0 ∧ − b_7_post ≤ 0 ∧ b_7_0 − b_7_post ≤ 0 ∧ − b_7_0 + b_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 1 3 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ b_7_0 ≤ 0 ∧ −1 + b_7_post ≤ 0 ∧ 1 − b_7_post ≤ 0 ∧ −1 − x_5_0 + x_5_post ≤ 0 ∧ 1 + x_5_0 − x_5_post ≤ 0 ∧ b_7_0 − b_7_post ≤ 0 ∧ − b_7_0 + b_7_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 ∧ − 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 4 2: 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 ∧ − b_7_post + b_7_post ≤ 0 ∧ b_7_post − b_7_post ≤ 0 ∧ − b_7_0 + b_7_0 ≤ 0 ∧ b_7_0 − b_7_0 ≤ 0 4 5 3: − 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 ∧ − b_7_post + b_7_post ≤ 0 ∧ b_7_post − b_7_post ≤ 0 ∧ − b_7_0 + b_7_0 ≤ 0 ∧ b_7_0 − b_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: −1 + b_7_post ≤ 0 ∧ 1 − b_7_post ≤ 0 ∧ −1 + b_7_0 ≤ 0 ∧ 1 − b_7_0 ≤ 0 1: b_7_post ≤ 0 ∧ − b_7_post ≤ 0 ∧ b_7_0 ≤ 0 ∧ − b_7_0 ≤ 0 2: − b_7_post ≤ 0 ∧ − b_7_0 ≤ 0 3: TRUE 4: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) −1 + b_7_post ≤ 0 ∧ 1 − b_7_post ≤ 0 ∧ −1 + b_7_0 ≤ 0 ∧ 1 − b_7_0 ≤ 0 1 (1) b_7_post ≤ 0 ∧ − b_7_post ≤ 0 ∧ b_7_0 ≤ 0 ∧ − b_7_0 ≤ 0 2 (2) − b_7_post ≤ 0 ∧ − b_7_0 ≤ 0 3 (3) TRUE 4 (4) TRUE
• initial node: 4
• cover edges:
• transition edges:  0 0 1 0 1 2 1 3 0 1 4 2 3 2 1 4 5 3

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 6 1: − 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 ∧ − b_7_post + b_7_post ≤ 0 ∧ b_7_post − b_7_post ≤ 0 ∧ − b_7_0 + b_7_0 ≤ 0 ∧ b_7_0 − b_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 1, 2, 4, 5 using the following ranking functions, which are bounded by −13.

 4: 0 3: 0 0: 0 1: 0 2: 0 4: −5 3: −6 0: −7 1: −7 1_var_snapshot: −7 1*: −7 2: −11

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

1* 9 1: 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 ≤ 0b_7_post + b_7_post ≤ 0b_7_postb_7_post ≤ 0b_7_0 + b_7_0 ≤ 0b_7_0b_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.

1 7 1_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 ≤ 0b_7_post + b_7_post ≤ 0b_7_postb_7_post ≤ 0b_7_0 + b_7_0 ≤ 0b_7_0b_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, 1, 1_var_snapshot, 1* }.

### 6.1.1 Transition Removal

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

 0: 3⋅b_7_post − 4⋅x_5_0 + 4⋅y_6_0 1: 1 − 4⋅x_5_0 + 4⋅y_6_0 1_var_snapshot: −4⋅x_5_0 + 4⋅y_6_0 1*: 2 − 4⋅x_5_0 + 4⋅y_6_0

### 6.1.2 Transition Removal

We remove transitions 7, 9 using the following ranking functions, which are bounded by −2.

 0: 0 1: −1 1_var_snapshot: −2 1*: 0

### 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 6.

### 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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