LTS Termination Proof

by T2Cert

Input

• Initial Location: 4
• Transitions: (pre-variables and post-variables)

  0	0	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg4P + arg4 ≤ 0 ∧ arg4P − arg4 ≤ 0 ∧ − arg5P + arg5 ≤ 0 ∧ arg5P − arg5 ≤ 0 ∧ − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0
  1	2	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −2 + arg1P − arg1 ≤ 0 ∧ 3 − arg1 + arg2P ≤ 0 ∧ 1 + arg2P − arg2 ≤ 0 ∧ 3 − arg1 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ 3 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg4P + arg4 ≤ 0 ∧ arg4P − arg4 ≤ 0 ∧ − arg5P + arg5 ≤ 0 ∧ arg5P − arg5 ≤ 0 ∧ − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0
  0	3	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ 1 − arg3P ≤ 0 ∧ −1 + arg1P − arg1 ≤ 0 ∧ −1 − arg1 + arg2P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 2 − arg1P ≤ 0 ∧ 2 − arg2P ≤ 0 ∧ arg2 − arg4P ≤ 0 ∧ − arg2 + arg4P ≤ 0 ∧ 1 − arg5P ≤ 0 ∧ −1 + arg5P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg4P + arg4 ≤ 0 ∧ arg4P − arg4 ≤ 0 ∧ − arg5P + arg5 ≤ 0 ∧ arg5P − arg5 ≤ 0 ∧ − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0
  3	4	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ − arg5 ≤ 0 ∧ 1 − arg4 + arg5 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 2 + arg2P − arg2 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 3 − arg2 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg2P ≤ 0 ∧ −1 − arg3P + arg3 ≤ 0 ∧ 1 + arg3P − arg3 ≤ 0 ∧ 1 − arg5P + arg5 ≤ 0 ∧ −1 + arg5P − arg5 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg5P + arg5 ≤ 0 ∧ arg5P − arg5 ≤ 0 ∧ − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0
  3	5	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg5 ≤ 0 ∧ 1 − x31 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ 1 − arg4 + arg5 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 2 + arg2P − arg2 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 3 − arg2 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg2P ≤ 0 ∧ −1 − arg3P + arg3 ≤ 0 ∧ 1 + arg3P − arg3 ≤ 0 ∧ 1 − arg5P + arg5 ≤ 0 ∧ −1 + arg5P − arg5 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg5P + arg5 ≤ 0 ∧ arg5P − arg5 ≤ 0 ∧ − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0
  3	6	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg5 ≤ 0 ∧ 1 − x39 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ 1 − arg4 + arg5 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 2 − arg2 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg2P ≤ 0 ∧ −1 − arg3P + arg3 ≤ 0 ∧ 1 + arg3P − arg3 ≤ 0 ∧ 1 − arg5P + arg5 ≤ 0 ∧ −1 + arg5P − arg5 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg5P + arg5 ≤ 0 ∧ arg5P − arg5 ≤ 0 ∧ − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0
  3	7	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ − arg5 ≤ 0 ∧ 1 − arg4 + arg5 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 2 − arg2 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ 1 − arg2P ≤ 0 ∧ −1 − arg3P + arg3 ≤ 0 ∧ 1 + arg3P − arg3 ≤ 0 ∧ 1 − arg5P + arg5 ≤ 0 ∧ −1 + arg5P − arg5 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg5P + arg5 ≤ 0 ∧ arg5P − arg5 ≤ 0 ∧ − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0
  3	8	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ − arg5 ≤ 0 ∧ 1 − arg4 + arg5 ≤ 0 ∧ −2 + arg1P − arg1 ≤ 0 ∧ −2 + arg1P − arg2 ≤ 0 ∧ −2 − arg1 + arg2P ≤ 0 ∧ −2 + arg2P − arg2 ≤ 0 ∧ 2 − arg1 ≤ 0 ∧ 2 − arg2 ≤ 0 ∧ 4 − arg1P ≤ 0 ∧ 4 − arg2P ≤ 0 ∧ −1 − arg3P + arg3 ≤ 0 ∧ 1 + arg3P − arg3 ≤ 0 ∧ 1 − arg5P + arg5 ≤ 0 ∧ −1 + arg5P − arg5 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg5P + arg5 ≤ 0 ∧ arg5P − arg5 ≤ 0 ∧ − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0
  3	9	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg5 ≤ 0 ∧ 1 − x61 ≤ 0 ∧ 1 − arg3 ≤ 0 ∧ 1 − arg4 + arg5 ≤ 0 ∧ −2 + arg1P − arg1 ≤ 0 ∧ −2 + arg1P − arg2 ≤ 0 ∧ −2 − arg1 + arg2P ≤ 0 ∧ −2 + arg2P − arg2 ≤ 0 ∧ 2 − arg1 ≤ 0 ∧ 2 − arg2 ≤ 0 ∧ 4 − arg1P ≤ 0 ∧ 4 − arg2P ≤ 0 ∧ −1 − arg3P + arg3 ≤ 0 ∧ 1 + arg3P − arg3 ≤ 0 ∧ 1 − arg5P + arg5 ≤ 0 ∧ −1 + arg5P − arg5 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg5P + arg5 ≤ 0 ∧ arg5P − arg5 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0
  4	10	0:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − arg4P + arg4 ≤ 0 ∧ arg4P − arg4 ≤ 0 ∧ − arg5P + arg5 ≤ 0 ∧ arg5P − arg5 ≤ 0 ∧ − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 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)		1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0
  3		(3)		− arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg1 ≤ 0 ∧ − arg2 ≤ 0
  4		(4)		TRUE

• initial node: 4
• cover edges:
• transition edges:

  0	0 1
  0	3 3
  1	2 1
  3	4 3
  3	5 3
  3	6 3
  3	7 3
  3	8 3
  3	9 3
  4	10 0

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 3, 10 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* 14 1: − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0 ∧ − arg5P + arg5P ≤ 0 ∧ arg5P − arg5P ≤ 0 ∧ − arg5 + arg5 ≤ 0 ∧ arg5 − arg5 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

5 Location Addition

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

1 12 1_var_snapshot: − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0 ∧ − arg5P + arg5P ≤ 0 ∧ arg5P − arg5P ≤ 0 ∧ − arg5 + arg5 ≤ 0 ∧ arg5 − arg5 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

6 Location Addition

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

3* 21 3: − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0 ∧ − arg5P + arg5P ≤ 0 ∧ arg5P − arg5P ≤ 0 ∧ − arg5 + arg5 ≤ 0 ∧ arg5 − arg5 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

7 Location Addition

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

3 19 3_var_snapshot: − x61 + x61 ≤ 0 ∧ x61 − x61 ≤ 0 ∧ − x39 + x39 ≤ 0 ∧ x39 − x39 ≤ 0 ∧ − x31 + x31 ≤ 0 ∧ x31 − x31 ≤ 0 ∧ − arg5P + arg5P ≤ 0 ∧ arg5P − arg5P ≤ 0 ∧ − arg5 + arg5 ≤ 0 ∧ arg5 − arg5 ≤ 0 ∧ − arg4P + arg4P ≤ 0 ∧ arg4P − arg4P ≤ 0 ∧ − arg4 + arg4 ≤ 0 ∧ arg4 − arg4 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

8 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

8.1 SCC Subproblem 1/2

Here we consider the SCC { 1, 1_var_snapshot, 1* }.

8.1.1 Transition Removal

We remove transitions 12, 14, 2 using the following ranking functions, which are bounded by −1.

8.1.2 Splitting Cut-Point Transitions

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

8.1.2.1 Cut-Point Subproblem 1/1

8.1.2.1.1 Splitting Cut-Point Transitions

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

8.2 SCC Subproblem 2/2

Here we consider the SCC { 3, 3_var_snapshot, 3* }.

8.2.1 Transition Removal

We remove transitions 4, 5, 6, 7, 8, 9 using the following ranking functions, which are bounded by 2.

8.2.2 Transition Removal

We remove transitions 19, 21 using the following ranking functions, which are bounded by −2.

8.2.3 Splitting Cut-Point Transitions

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

8.2.3.1 Cut-Point Subproblem 1/1

8.2.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

• version: 1.0