# 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 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_13_1 − s_17_0 ≤ 0 ∧ − i_13_1 + s_17_0 ≤ 0 ∧ j_15_post − s_16_0 ≤ 0 ∧ − j_15_post + s_16_0 ≤ 0 ∧ i_13_post − j_15_post ≤ 0 ∧ − i_13_post + j_15_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ j_15_0 − j_15_post ≤ 0 ∧ − j_15_0 + j_15_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 2 1 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 500 − i_13_0 ≤ 0 ∧ rt_11_post − st_14_0 ≤ 0 ∧ − rt_11_post + st_14_0 ≤ 0 ∧ rt_11_0 − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 2 2 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −499 + i_13_0 ≤ 0 ∧ −1 − i_13_0 + i_13_post ≤ 0 ∧ 1 + i_13_0 − i_13_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 4 3 2: − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 1 4 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 500 − i_13_0 ≤ 0 ∧ rt_11_post − st_14_0 ≤ 0 ∧ − rt_11_post + st_14_0 ≤ 0 ∧ rt_11_0 − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 1 5 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −499 + i_13_0 ≤ 0 ∧ −1 − i_13_0 + i_13_post ≤ 0 ∧ 1 + i_13_0 − i_13_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 5 6 0: − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0

## Proof

The following invariants are asserted.

 0: TRUE 1: TRUE 2: TRUE 3: 500 − i_13_0 ≤ 0 4: TRUE 5: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) TRUE 2 (2) TRUE 3 (3) 500 − i_13_0 ≤ 0 4 (4) TRUE 5 (5) TRUE
• initial node: 5
• cover edges:
• transition edges:  0 0 1 1 4 3 1 5 2 2 1 3 2 2 4 4 3 2 5 6 0

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 2 7 2: − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 1, 4, 5, 6 using the following ranking functions, which are bounded by −15.

 5: 0 0: 0 1: 0 2: 0 4: 0 3: 0 5: −6 0: −7 1: −8 2: −9 4: −9 2_var_snapshot: −9 2*: −9 3: −13

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

2* 10 2: st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0s_17_0 + s_17_0 ≤ 0s_17_0s_17_0 ≤ 0s_16_0 + s_16_0 ≤ 0s_16_0s_16_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0j_15_post + j_15_post ≤ 0j_15_postj_15_post ≤ 0j_15_0 + j_15_0 ≤ 0j_15_0j_15_0 ≤ 0i_13_post + i_13_post ≤ 0i_13_posti_13_post ≤ 0i_13_1 + i_13_1 ≤ 0i_13_1i_13_1 ≤ 0i_13_0 + i_13_0 ≤ 0i_13_0i_13_0 ≤ 0

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

2 8 2_var_snapshot: st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0s_17_0 + s_17_0 ≤ 0s_17_0s_17_0 ≤ 0s_16_0 + s_16_0 ≤ 0s_16_0s_16_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0j_15_post + j_15_post ≤ 0j_15_postj_15_post ≤ 0j_15_0 + j_15_0 ≤ 0j_15_0j_15_0 ≤ 0i_13_post + i_13_post ≤ 0i_13_posti_13_post ≤ 0i_13_1 + i_13_1 ≤ 0i_13_1i_13_1 ≤ 0i_13_0 + i_13_0 ≤ 0i_13_0i_13_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 { 2, 4, 2_var_snapshot, 2* }.

### 6.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −2000.

 2: −2 − 4⋅i_13_0 4: −4⋅i_13_0 2_var_snapshot: −3 − 4⋅i_13_0 2*: −1 − 4⋅i_13_0

### 6.1.2 Transition Removal

We remove transitions 8, 10 using the following ranking functions, which are bounded by −1.

 2: 0 4: 2 2_var_snapshot: −1 2*: 1

### 6.1.3 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −1.

 2: 0 4: 0 2_var_snapshot: 0 2*: −1

### 6.1.4 Splitting Cut-Point Transitions

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

### 6.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

### 6.1.4.1.1 Splitting Cut-Point Transitions

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

T2Cert

• version: 1.0