LTS Termination Proof

by T2Cert

Input

• 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.

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

3 Transition Removal

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

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.

6.1.2 Transition Removal

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

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

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

T2Cert

• version: 1.0