LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 4

Transitions: (pre-variables and post-variables)

0	0	1:	− y_21_post + y_21_post ≤ 0 ∧ y_21_post − y_21_post ≤ 0 ∧ − y_21_0 + y_21_0 ≤ 0 ∧ y_21_0 − y_21_0 ≤ 0 ∧ − y_15_post + y_15_post ≤ 0 ∧ y_15_post − y_15_post ≤ 0 ∧ − y_15_1 + y_15_1 ≤ 0 ∧ y_15_1 − y_15_1 ≤ 0 ∧ − y_15_0 + y_15_0 ≤ 0 ∧ y_15_0 − y_15_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_1 + x_13_1 ≤ 0 ∧ x_13_1 − x_13_1 ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − t_22_post + t_22_post ≤ 0 ∧ t_22_post − t_22_post ≤ 0 ∧ − t_22_0 + t_22_0 ≤ 0 ∧ t_22_0 − t_22_0 ≤ 0 ∧ − t_16_post + t_16_post ≤ 0 ∧ t_16_post − t_16_post ≤ 0 ∧ − t_16_1 + t_16_1 ≤ 0 ∧ t_16_1 − t_16_1 ≤ 0 ∧ − t_16_0 + t_16_0 ≤ 0 ∧ t_16_0 − t_16_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0
1	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_13_0 ≤ 0 ∧ y_15_0 ≤ 0 ∧ result_11_post − temp0_14_0 ≤ 0 ∧ − result_11_post + temp0_14_0 ≤ 0 ∧ result_11_0 − result_11_post ≤ 0 ∧ − result_11_0 + result_11_post ≤ 0 ∧ − y_21_post + y_21_post ≤ 0 ∧ y_21_post − y_21_post ≤ 0 ∧ − y_21_0 + y_21_0 ≤ 0 ∧ y_21_0 − y_21_0 ≤ 0 ∧ − y_15_post + y_15_post ≤ 0 ∧ y_15_post − y_15_post ≤ 0 ∧ − y_15_1 + y_15_1 ≤ 0 ∧ y_15_1 − y_15_1 ≤ 0 ∧ − y_15_0 + y_15_0 ≤ 0 ∧ y_15_0 − y_15_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_1 + x_13_1 ≤ 0 ∧ x_13_1 − x_13_1 ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − t_22_post + t_22_post ≤ 0 ∧ t_22_post − t_22_post ≤ 0 ∧ − t_22_0 + t_22_0 ≤ 0 ∧ t_22_0 − t_22_0 ≤ 0 ∧ − t_16_post + t_16_post ≤ 0 ∧ t_16_post − t_16_post ≤ 0 ∧ − t_16_1 + t_16_1 ≤ 0 ∧ t_16_1 − t_16_1 ≤ 0 ∧ − t_16_0 + t_16_0 ≤ 0 ∧ t_16_0 − t_16_0 ≤ 0
1	2	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ x_13_0 ≤ 0 ∧ result_11_post − temp0_14_0 ≤ 0 ∧ − result_11_post + temp0_14_0 ≤ 0 ∧ x_13_post ≤ 0 ∧ result_11_0 − result_11_post ≤ 0 ∧ − result_11_0 + result_11_post ≤ 0 ∧ x_13_0 − x_13_post ≤ 0 ∧ − x_13_0 + x_13_post ≤ 0 ∧ − y_21_post + y_21_post ≤ 0 ∧ y_21_post − y_21_post ≤ 0 ∧ − y_21_0 + y_21_0 ≤ 0 ∧ y_21_0 − y_21_0 ≤ 0 ∧ − y_15_post + y_15_post ≤ 0 ∧ y_15_post − y_15_post ≤ 0 ∧ − y_15_1 + y_15_1 ≤ 0 ∧ y_15_1 − y_15_1 ≤ 0 ∧ − y_15_0 + y_15_0 ≤ 0 ∧ y_15_0 − y_15_0 ≤ 0 ∧ − x_13_1 + x_13_1 ≤ 0 ∧ x_13_1 − x_13_1 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − t_22_post + t_22_post ≤ 0 ∧ t_22_post − t_22_post ≤ 0 ∧ − t_22_0 + t_22_0 ≤ 0 ∧ t_22_0 − t_22_0 ≤ 0 ∧ − t_16_post + t_16_post ≤ 0 ∧ t_16_post − t_16_post ≤ 0 ∧ − t_16_1 + t_16_1 ≤ 0 ∧ t_16_1 − t_16_1 ≤ 0 ∧ − t_16_0 + t_16_0 ≤ 0 ∧ t_16_0 − t_16_0 ≤ 0
1	3	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 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 ∧ 1 − y_15_0 ≤ 0 ∧ t_16_1 − x_13_0 ≤ 0 ∧ − t_16_1 + x_13_0 ≤ 0 ∧ 2 + x_13_1 − y_15_0 ≤ 0 ∧ −2 − x_13_1 + y_15_0 ≤ 0 ∧ −1 − t_16_1 + y_15_1 ≤ 0 ∧ 1 + t_16_1 − y_15_1 ≤ 0 ∧ t_16_post − x_13_1 ≤ 0 ∧ − t_16_post + x_13_1 ≤ 0 ∧ 2 + x_13_post − y_15_1 ≤ 0 ∧ −2 − x_13_post + y_15_1 ≤ 0 ∧ −1 − t_16_post + y_15_post ≤ 0 ∧ 1 + t_16_post − y_15_post ≤ 0 ∧ 1 − t_22_post + x_13_post ≤ 0 ∧ −1 + t_22_post − x_13_post ≤ 0 ∧ −1 − t_16_post + y_15_post ≤ 0 ∧ 1 + t_16_post − y_15_post ≤ 0 ∧ 2 + t_16_post − y_21_post ≤ 0 ∧ −2 − t_16_post + y_21_post ≤ 0 ∧ 1 − y_21_post ≤ 0 ∧ 1 − t_22_post ≤ 0 ∧ t_16_0 − t_16_post ≤ 0 ∧ − t_16_0 + t_16_post ≤ 0 ∧ t_22_0 − t_22_post ≤ 0 ∧ − t_22_0 + t_22_post ≤ 0 ∧ x_13_0 − x_13_post ≤ 0 ∧ − x_13_0 + x_13_post ≤ 0 ∧ y_15_0 − y_15_post ≤ 0 ∧ − y_15_0 + y_15_post ≤ 0 ∧ y_21_0 − y_21_post ≤ 0 ∧ − y_21_0 + y_21_post ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0
3	4	1:	− y_21_post + y_21_post ≤ 0 ∧ y_21_post − y_21_post ≤ 0 ∧ − y_21_0 + y_21_0 ≤ 0 ∧ y_21_0 − y_21_0 ≤ 0 ∧ − y_15_post + y_15_post ≤ 0 ∧ y_15_post − y_15_post ≤ 0 ∧ − y_15_1 + y_15_1 ≤ 0 ∧ y_15_1 − y_15_1 ≤ 0 ∧ − y_15_0 + y_15_0 ≤ 0 ∧ y_15_0 − y_15_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_1 + x_13_1 ≤ 0 ∧ x_13_1 − x_13_1 ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − t_22_post + t_22_post ≤ 0 ∧ t_22_post − t_22_post ≤ 0 ∧ − t_22_0 + t_22_0 ≤ 0 ∧ t_22_0 − t_22_0 ≤ 0 ∧ − t_16_post + t_16_post ≤ 0 ∧ t_16_post − t_16_post ≤ 0 ∧ − t_16_1 + t_16_1 ≤ 0 ∧ t_16_1 − t_16_1 ≤ 0 ∧ − t_16_0 + t_16_0 ≤ 0 ∧ t_16_0 − t_16_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0
4	5	0:	− y_21_post + y_21_post ≤ 0 ∧ y_21_post − y_21_post ≤ 0 ∧ − y_21_0 + y_21_0 ≤ 0 ∧ y_21_0 − y_21_0 ≤ 0 ∧ − y_15_post + y_15_post ≤ 0 ∧ y_15_post − y_15_post ≤ 0 ∧ − y_15_1 + y_15_1 ≤ 0 ∧ y_15_1 − y_15_1 ≤ 0 ∧ − y_15_0 + y_15_0 ≤ 0 ∧ y_15_0 − y_15_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_1 + x_13_1 ≤ 0 ∧ x_13_1 − x_13_1 ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − t_22_post + t_22_post ≤ 0 ∧ t_22_post − t_22_post ≤ 0 ∧ − t_22_0 + t_22_0 ≤ 0 ∧ t_22_0 − t_22_0 ≤ 0 ∧ − t_16_post + t_16_post ≤ 0 ∧ t_16_post − t_16_post ≤ 0 ∧ − t_16_1 + t_16_1 ≤ 0 ∧ t_16_1 − t_16_1 ≤ 0 ∧ − t_16_0 + t_16_0 ≤ 0 ∧ t_16_0 − t_16_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	TRUE
2:	TRUE
3:	1 − t_22_post ≤ 0 ∧ 1 − y_21_post ≤ 0 ∧ 1 − t_22_0 ≤ 0 ∧ 1 − y_21_0 ≤ 0
4:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	TRUE
2	(2)	TRUE
3	(3)	1 − t_22_post ≤ 0 ∧ 1 − y_21_post ≤ 0 ∧ 1 − t_22_0 ≤ 0 ∧ 1 − y_21_0 ≤ 0
4	(4)	TRUE

initial node: 4
cover edges:
transition edges:

0 0 1

1 1 2

1 2 2

1 3 3

3 4 1

4 5 0

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− y_21_post + y_21_post ≤ 0 ∧ y_21_post − y_21_post ≤ 0 ∧ − y_21_0 + y_21_0 ≤ 0 ∧ y_21_0 − y_21_0 ≤ 0 ∧ − y_15_post + y_15_post ≤ 0 ∧ y_15_post − y_15_post ≤ 0 ∧ − y_15_1 + y_15_1 ≤ 0 ∧ y_15_1 − y_15_1 ≤ 0 ∧ − y_15_0 + y_15_0 ≤ 0 ∧ y_15_0 − y_15_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_1 + x_13_1 ≤ 0 ∧ x_13_1 − x_13_1 ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − t_22_post + t_22_post ≤ 0 ∧ t_22_post − t_22_post ≤ 0 ∧ − t_22_0 + t_22_0 ≤ 0 ∧ t_22_0 − t_22_0 ≤ 0 ∧ − t_16_post + t_16_post ≤ 0 ∧ t_16_post − t_16_post ≤ 0 ∧ − t_16_1 + t_16_1 ≤ 0 ∧ t_16_1 − t_16_1 ≤ 0 ∧ − t_16_0 + t_16_0 ≤ 0 ∧ t_16_0 − t_16_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0

and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 1, 2, 5 using the following ranking functions, which are bounded by −13.

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

4 Location Addition

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

1* 9 1: − y_21_post + y_21_post ≤ 0 ∧ y_21_post − y_21_post ≤ 0 ∧ − y_21_0 + y_21_0 ≤ 0 ∧ y_21_0 − y_21_0 ≤ 0 ∧ − y_15_post + y_15_post ≤ 0 ∧ y_15_post − y_15_post ≤ 0 ∧ − y_15_1 + y_15_1 ≤ 0 ∧ y_15_1 − y_15_1 ≤ 0 ∧ − y_15_0 + y_15_0 ≤ 0 ∧ y_15_0 − y_15_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_1 + x_13_1 ≤ 0 ∧ x_13_1 − x_13_1 ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − t_22_post + t_22_post ≤ 0 ∧ t_22_post − t_22_post ≤ 0 ∧ − t_22_0 + t_22_0 ≤ 0 ∧ t_22_0 − t_22_0 ≤ 0 ∧ − t_16_post + t_16_post ≤ 0 ∧ t_16_post − t_16_post ≤ 0 ∧ − t_16_1 + t_16_1 ≤ 0 ∧ t_16_1 − t_16_1 ≤ 0 ∧ − t_16_0 + t_16_0 ≤ 0 ∧ t_16_0 − t_16_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0

5 Location Addition

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

1 7 1_var_snapshot: − y_21_post + y_21_post ≤ 0 ∧ y_21_post − y_21_post ≤ 0 ∧ − y_21_0 + y_21_0 ≤ 0 ∧ y_21_0 − y_21_0 ≤ 0 ∧ − y_15_post + y_15_post ≤ 0 ∧ y_15_post − y_15_post ≤ 0 ∧ − y_15_1 + y_15_1 ≤ 0 ∧ y_15_1 − y_15_1 ≤ 0 ∧ − y_15_0 + y_15_0 ≤ 0 ∧ y_15_0 − y_15_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_1 + x_13_1 ≤ 0 ∧ x_13_1 − x_13_1 ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − t_22_post + t_22_post ≤ 0 ∧ t_22_post − t_22_post ≤ 0 ∧ − t_22_0 + t_22_0 ≤ 0 ∧ t_22_0 − t_22_0 ≤ 0 ∧ − t_16_post + t_16_post ≤ 0 ∧ t_16_post − t_16_post ≤ 0 ∧ − t_16_1 + t_16_1 ≤ 0 ∧ t_16_1 − t_16_1 ≤ 0 ∧ − t_16_0 + t_16_0 ≤ 0 ∧ t_16_0 − t_16_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_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, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

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

1:	−2 + x_13_0 + 3⋅y_15_0
3:	x_13_0 + 3⋅y_15_0
1_var_snapshot:	−3 + x_13_0 + 3⋅y_15_0
1*:	−1 + x_13_0 + 3⋅y_15_0

6.1.2 Transition Removal

We remove transitions 7, 9, 4 using the following ranking functions, which are bounded by −2.

1:	−1
3:	y_21_0
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.

Tool configuration

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