# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 7
• Transitions: (pre-variables and post-variables)  0 0 1: − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_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 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_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 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0 1 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ x_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 ∧ − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_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 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_0 ≤ 0 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0 1 2 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ − nd_12_1 + rv_15_post ≤ 0 ∧ nd_12_1 − rv_15_post ≤ 0 ∧ − rv_15_post ≤ 0 ∧ rv_15_post ≤ 0 ∧ 1 − y_17_0 + y_17_post ≤ 0 ∧ −1 + y_17_0 − y_17_post ≤ 0 ∧ 2 − y_17_post ≤ 0 ∧ nd_12_0 − nd_12_post ≤ 0 ∧ − nd_12_0 + nd_12_post ≤ 0 ∧ rv_15_0 − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_post ≤ 0 ∧ st_16_0 − st_16_post ≤ 0 ∧ − st_16_0 + st_16_post ≤ 0 ∧ y_17_0 − y_17_post ≤ 0 ∧ − y_17_0 + y_17_post ≤ 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 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_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 3 3 1: − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_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 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_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 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0 1 4 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ − nd_12_1 + rv_15_post ≤ 0 ∧ nd_12_1 − rv_15_post ≤ 0 ∧ nd_12_0 − nd_12_post ≤ 0 ∧ − nd_12_0 + nd_12_post ≤ 0 ∧ rv_15_0 − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_post ≤ 0 ∧ − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_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 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_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 4 5 5: − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_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 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_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 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0 5 6 6: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_13_0 + x_13_post ≤ 0 ∧ −1 + x_13_0 − x_13_post ≤ 0 ∧ − nd_12_1 + y_17_post ≤ 0 ∧ nd_12_1 − y_17_post ≤ 0 ∧ nd_12_0 − nd_12_post ≤ 0 ∧ − nd_12_0 + nd_12_post ≤ 0 ∧ x_13_0 − x_13_post ≤ 0 ∧ − x_13_0 + x_13_post ≤ 0 ∧ y_17_0 − y_17_post ≤ 0 ∧ − y_17_0 + y_17_post ≤ 0 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_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 6 7 1: − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_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 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_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 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0 7 8 0: − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_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 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_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 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0

## Proof

The following invariants are asserted.

 0: TRUE 1: TRUE 2: x_13_0 ≤ 0 3: rv_15_post ≤ 0 ∧ − rv_15_post ≤ 0 ∧ 2 − y_17_post ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ 2 − y_17_0 ≤ 0 ∧ rv_15_0 ≤ 0 ∧ − rv_15_0 ≤ 0 4: 1 − x_13_0 ≤ 0 5: 1 − x_13_0 ≤ 0 6: TRUE 7: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) TRUE 2 (2) x_13_0 ≤ 0 3 (3) rv_15_post ≤ 0 ∧ − rv_15_post ≤ 0 ∧ 2 − y_17_post ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ 2 − y_17_0 ≤ 0 ∧ rv_15_0 ≤ 0 ∧ − rv_15_0 ≤ 0 4 (4) 1 − x_13_0 ≤ 0 5 (5) 1 − x_13_0 ≤ 0 6 (6) TRUE 7 (7) TRUE
• initial node: 7
• cover edges:
• transition edges:  0 0 1 1 1 2 1 2 3 1 4 4 3 3 1 4 5 5 5 6 6 6 7 1 7 8 0

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 9 1: − y_17_post + y_17_post ≤ 0 ∧ y_17_post − y_17_post ≤ 0 ∧ − y_17_0 + y_17_0 ≤ 0 ∧ y_17_0 − y_17_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 ∧ − st_16_post + st_16_post ≤ 0 ∧ st_16_post − st_16_post ≤ 0 ∧ − st_16_0 + st_16_0 ≤ 0 ∧ st_16_0 − st_16_0 ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rv_15_post + rv_15_post ≤ 0 ∧ rv_15_post − rv_15_post ≤ 0 ∧ − rv_15_0 + rv_15_0 ≤ 0 ∧ rv_15_0 − rv_15_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 ∧ − nd_12_post + nd_12_post ≤ 0 ∧ nd_12_post − nd_12_post ≤ 0 ∧ − nd_12_1 + nd_12_1 ≤ 0 ∧ nd_12_1 − nd_12_1 ≤ 0 ∧ − nd_12_0 + nd_12_0 ≤ 0 ∧ nd_12_0 − nd_12_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 1, 8 using the following ranking functions, which are bounded by −13.

 7: 0 0: 0 1: 0 3: 0 4: 0 5: 0 6: 0 2: 0 7: −5 0: −6 1: −7 3: −7 4: −7 5: −7 6: −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* 12 1: y_17_post + y_17_post ≤ 0y_17_posty_17_post ≤ 0y_17_0 + y_17_0 ≤ 0y_17_0y_17_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0st_16_post + st_16_post ≤ 0st_16_postst_16_post ≤ 0st_16_0 + st_16_0 ≤ 0st_16_0st_16_0 ≤ 0st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rv_15_post + rv_15_post ≤ 0rv_15_postrv_15_post ≤ 0rv_15_0 + rv_15_0 ≤ 0rv_15_0rv_15_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0nd_12_post + nd_12_post ≤ 0nd_12_postnd_12_post ≤ 0nd_12_1 + nd_12_1 ≤ 0nd_12_1nd_12_1 ≤ 0nd_12_0 + nd_12_0 ≤ 0nd_12_0nd_12_0 ≤ 0

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

1 10 1_var_snapshot: y_17_post + y_17_post ≤ 0y_17_posty_17_post ≤ 0y_17_0 + y_17_0 ≤ 0y_17_0y_17_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0st_16_post + st_16_post ≤ 0st_16_postst_16_post ≤ 0st_16_0 + st_16_0 ≤ 0st_16_0st_16_0 ≤ 0st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rv_15_post + rv_15_post ≤ 0rv_15_postrv_15_post ≤ 0rv_15_0 + rv_15_0 ≤ 0rv_15_0rv_15_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0nd_12_post + nd_12_post ≤ 0nd_12_postnd_12_post ≤ 0nd_12_1 + nd_12_1 ≤ 0nd_12_1nd_12_1 ≤ 0nd_12_0 + nd_12_0 ≤ 0nd_12_0nd_12_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 { 1, 3, 4, 5, 6, 1_var_snapshot, 1* }.

### 6.1.1 Transition Removal

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

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

### 6.1.2 Transition Removal

We remove transitions 4, 7 using the following ranking functions, which are bounded by −2.

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

### 6.1.3 Transition Removal

We remove transitions 2, 3 using the following ranking functions, which are bounded by 8.

 1: −1 + 4⋅y_17_0 3: 1 + 4⋅y_17_0 4: 0 5: 0 6: 0 1_var_snapshot: −2 + 4⋅y_17_0 1*: 4⋅y_17_0

### 6.1.4 Transition Removal

We remove transitions 10, 12 using the following ranking functions, which are bounded by −2.

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

### 6.1.5 Splitting Cut-Point Transitions

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

### 6.1.5.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 9.

### 6.1.5.1.1 Splitting Cut-Point Transitions

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

T2Cert

• version: 1.0