LTS Termination Proof

by T2Cert

Input

• Initial Location: 6
• 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 ∧ − 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
  1	1	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_13_0 ≤ 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 ∧ − 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
  1	2	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ x_13_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_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 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0
  2	4	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 ∧ − 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
  2	5	5:		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 ∧ − 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
  5	6	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 ∧ − 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
  6	7	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 ∧ − 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

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)		x_13_0 ≤ 0
  5		(5)		1 − y_33_post ≤ 0 ∧ 1 − y_33_0 ≤ 0
  6		(6)		TRUE

• initial node: 6
• cover edges:
• transition edges:

  0	0 1
  1	1 2
  1	2 3
  2	4 1
  2	5 5
  5	6 2
  6	7 0

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 2, 7 using the following ranking functions, which are bounded by −15.

4 Location Addition

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

1* 11 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 ∧ − 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

5 Location Addition

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

1 9 1_var_snapshot: − 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 ∧ − 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

6 Location Addition

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

2* 18 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 ∧ − 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

7 Location Addition

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

2 16 2_var_snapshot: − 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 ∧ − 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

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, 5, 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.

8.1.2 Transition Removal

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

8.1.3 Transition Removal

We remove transitions 11, 16, 18, 4, 6 using the following ranking functions, which are bounded by −3.

8.1.4 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by −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

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

8.1.5.2.1 Splitting Cut-Point Transitions

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

Tool configuration

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