# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 5
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ x_13_0 − x_13_post ≤ 0 ∧ − x_13_0 + x_13_post ≤ 0 ∧ y_16_0 − y_16_post ≤ 0 ∧ − y_16_0 + y_16_post ≤ 0 ∧ − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − ___const_5000_0 + ___const_5000_0 ≤ 0 ∧ ___const_5000_0 − ___const_5000_0 ≤ 0 1 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ − ___const_5000_0 + y_16_post ≤ 0 ∧ ___const_5000_0 − y_16_post ≤ 0 ∧ 1 − y_16_post ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ y_16_0 − y_16_post ≤ 0 ∧ − y_16_0 + y_16_post ≤ 0 ∧ − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − ___const_5000_0 + ___const_5000_0 ≤ 0 ∧ ___const_5000_0 − ___const_5000_0 ≤ 0 2 3 1: y_16_0 ≤ 0 ∧ y_16_0 ≤ 0 ∧ − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − ___const_5000_0 + ___const_5000_0 ≤ 0 ∧ ___const_5000_0 − ___const_5000_0 ≤ 0 2 4 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − y_16_0 ≤ 0 ∧ 1 − x_13_0 + x_13_post ≤ 0 ∧ −1 + x_13_0 − x_13_post ≤ 0 ∧ 1 − y_16_0 + y_16_post ≤ 0 ∧ −1 + y_16_0 − y_16_post ≤ 0 ∧ 1 + x_13_post − x_32_post ≤ 0 ∧ −1 − x_13_post + x_32_post ≤ 0 ∧ 1 + y_16_post − y_33_post ≤ 0 ∧ −1 − y_16_post + y_33_post ≤ 0 ∧ 1 − y_33_post ≤ 0 ∧ x_13_0 − x_13_post ≤ 0 ∧ − x_13_0 + x_13_post ≤ 0 ∧ x_32_0 − x_32_post ≤ 0 ∧ − x_32_0 + x_32_post ≤ 0 ∧ y_16_0 − y_16_post ≤ 0 ∧ − y_16_0 + y_16_post ≤ 0 ∧ y_33_0 − y_33_post ≤ 0 ∧ − y_33_0 + y_33_post ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − ___const_5000_0 + ___const_5000_0 ≤ 0 ∧ ___const_5000_0 − ___const_5000_0 ≤ 0 4 5 2: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − ___const_5000_0 + ___const_5000_0 ≤ 0 ∧ ___const_5000_0 − ___const_5000_0 ≤ 0 5 6 0: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − ___const_5000_0 + ___const_5000_0 ≤ 0 ∧ ___const_5000_0 − ___const_5000_0 ≤ 0

## Proof

The following invariants are asserted.

 0: TRUE 1: TRUE 2: TRUE 4: 1 − y_33_post ≤ 0 ∧ 1 − y_33_0 ≤ 0 5: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) TRUE 2 (2) TRUE 4 (4) 1 − y_33_post ≤ 0 ∧ 1 − y_33_0 ≤ 0 5 (5) TRUE
• initial node: 5
• cover edges:
• transition edges:  0 0 1 1 1 2 2 3 1 2 4 4 4 5 2 5 6 0

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 7 1: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − ___const_5000_0 + ___const_5000_0 ≤ 0 ∧ ___const_5000_0 − ___const_5000_0 ≤ 0 2 14 2: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − ___const_5000_0 + ___const_5000_0 ≤ 0 ∧ ___const_5000_0 − ___const_5000_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 6 using the following ranking functions, which are bounded by −13.

 5: 0 0: 0 1: 0 2: 0 4: 0 5: −4 0: −5 1: −6 2: −6 4: −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* 10 1: y_33_post + y_33_post ≤ 0y_33_posty_33_post ≤ 0y_33_0 + y_33_0 ≤ 0y_33_0y_33_0 ≤ 0y_28_post + y_28_post ≤ 0y_28_posty_28_post ≤ 0y_28_0 + y_28_0 ≤ 0y_28_0y_28_0 ≤ 0y_16_post + y_16_post ≤ 0y_16_posty_16_post ≤ 0y_16_0 + y_16_0 ≤ 0y_16_0y_16_0 ≤ 0x_32_post + x_32_post ≤ 0x_32_postx_32_post ≤ 0x_32_0 + x_32_0 ≤ 0x_32_0x_32_0 ≤ 0x_27_post + x_27_post ≤ 0x_27_postx_27_post ≤ 0x_27_0 + x_27_0 ≤ 0x_27_0x_27_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0___const_5000_0 + ___const_5000_0 ≤ 0___const_5000_0___const_5000_0 ≤ 0

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

1 8 1_var_snapshot: y_33_post + y_33_post ≤ 0y_33_posty_33_post ≤ 0y_33_0 + y_33_0 ≤ 0y_33_0y_33_0 ≤ 0y_28_post + y_28_post ≤ 0y_28_posty_28_post ≤ 0y_28_0 + y_28_0 ≤ 0y_28_0y_28_0 ≤ 0y_16_post + y_16_post ≤ 0y_16_posty_16_post ≤ 0y_16_0 + y_16_0 ≤ 0y_16_0y_16_0 ≤ 0x_32_post + x_32_post ≤ 0x_32_postx_32_post ≤ 0x_32_0 + x_32_0 ≤ 0x_32_0x_32_0 ≤ 0x_27_post + x_27_post ≤ 0x_27_postx_27_post ≤ 0x_27_0 + x_27_0 ≤ 0x_27_0x_27_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0___const_5000_0 + ___const_5000_0 ≤ 0___const_5000_0___const_5000_0 ≤ 0

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

2* 17 2: y_33_post + y_33_post ≤ 0y_33_posty_33_post ≤ 0y_33_0 + y_33_0 ≤ 0y_33_0y_33_0 ≤ 0y_28_post + y_28_post ≤ 0y_28_posty_28_post ≤ 0y_28_0 + y_28_0 ≤ 0y_28_0y_28_0 ≤ 0y_16_post + y_16_post ≤ 0y_16_posty_16_post ≤ 0y_16_0 + y_16_0 ≤ 0y_16_0y_16_0 ≤ 0x_32_post + x_32_post ≤ 0x_32_postx_32_post ≤ 0x_32_0 + x_32_0 ≤ 0x_32_0x_32_0 ≤ 0x_27_post + x_27_post ≤ 0x_27_postx_27_post ≤ 0x_27_0 + x_27_0 ≤ 0x_27_0x_27_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0___const_5000_0 + ___const_5000_0 ≤ 0___const_5000_0___const_5000_0 ≤ 0

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

2 15 2_var_snapshot: y_33_post + y_33_post ≤ 0y_33_posty_33_post ≤ 0y_33_0 + y_33_0 ≤ 0y_33_0y_33_0 ≤ 0y_28_post + y_28_post ≤ 0y_28_posty_28_post ≤ 0y_28_0 + y_28_0 ≤ 0y_28_0y_28_0 ≤ 0y_16_post + y_16_post ≤ 0y_16_posty_16_post ≤ 0y_16_0 + y_16_0 ≤ 0y_16_0y_16_0 ≤ 0x_32_post + x_32_post ≤ 0x_32_postx_32_post ≤ 0x_32_0 + x_32_0 ≤ 0x_32_0x_32_0 ≤ 0x_27_post + x_27_post ≤ 0x_27_postx_27_post ≤ 0x_27_0 + x_27_0 ≤ 0x_27_0x_27_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0___const_5000_0 + ___const_5000_0 ≤ 0___const_5000_0___const_5000_0 ≤ 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, 4, 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 4.

 1: −4 + 10⋅x_13_0 2: −1 + 10⋅x_13_0 − 6⋅y_16_0 4: 1 + 10⋅x_13_0 − 6⋅y_16_0 1_var_snapshot: −5 + 10⋅x_13_0 1*: −3 + 10⋅x_13_0 2_var_snapshot: −2 + 10⋅x_13_0 − 6⋅y_16_0 2*: 10⋅x_13_0 − 6⋅y_16_0

### 8.1.2 Transition Removal

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

 1: −3 + 5⋅y_16_0 2: −1 + 5⋅y_16_0 4: 1 + 5⋅y_16_0 1_var_snapshot: −4 + 5⋅y_16_0 1*: −2 + 5⋅y_16_0 2_var_snapshot: −2 + 5⋅y_16_0 2*: 5⋅y_16_0

### 8.1.3 Transition Removal

We remove transitions 8, 10, 15, 17, 3 using the following ranking functions, which are bounded by −3.

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

### 8.1.4 Transition Removal

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

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

### 8.1.5 Splitting Cut-Point Transitions

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

### 8.1.5.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 7.

### 8.1.5.1.1 Splitting Cut-Point Transitions

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

### 8.1.5.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 14.

### 8.1.5.2.1 Splitting Cut-Point Transitions

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

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