LTS Termination Proof

by T2Cert

Input

• Initial Location: 6
• Transitions: (pre-variables and post-variables)

  0	0	1:		3 − a_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − ret_returnOne3_post + ret_returnOne3_post ≤ 0 ∧ ret_returnOne3_post − ret_returnOne3_post ≤ 0 ∧ − ret_returnOne3_0 + ret_returnOne3_0 ≤ 0 ∧ ret_returnOne3_0 − ret_returnOne3_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_1 + a_1 ≤ 0 ∧ a_1 − a_1 ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  0	1	1:		−1 + a_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − ret_returnOne3_post + ret_returnOne3_post ≤ 0 ∧ ret_returnOne3_post − ret_returnOne3_post ≤ 0 ∧ − ret_returnOne3_0 + ret_returnOne3_0 ≤ 0 ∧ ret_returnOne3_0 − ret_returnOne3_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_1 + a_1 ≤ 0 ∧ a_1 − a_1 ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  0	2	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −2 + a_0 ≤ 0 ∧ 2 − a_0 ≤ 0 ∧ −1 + tmp_post ≤ 0 ∧ 1 − tmp_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − ret_returnOne3_post + ret_returnOne3_post ≤ 0 ∧ ret_returnOne3_post − ret_returnOne3_post ≤ 0 ∧ − ret_returnOne3_0 + ret_returnOne3_0 ≤ 0 ∧ ret_returnOne3_0 − ret_returnOne3_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_1 + a_1 ≤ 0 ∧ a_1 − a_1 ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  2	3	3:		− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − ret_returnOne3_post + ret_returnOne3_post ≤ 0 ∧ ret_returnOne3_post − ret_returnOne3_post ≤ 0 ∧ − ret_returnOne3_0 + ret_returnOne3_0 ≤ 0 ∧ ret_returnOne3_0 − ret_returnOne3_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_1 + a_1 ≤ 0 ∧ a_1 − a_1 ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  4	4	0:		2 − a_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − ret_returnOne3_post + ret_returnOne3_post ≤ 0 ∧ ret_returnOne3_post − ret_returnOne3_post ≤ 0 ∧ − ret_returnOne3_0 + ret_returnOne3_0 ≤ 0 ∧ ret_returnOne3_0 − ret_returnOne3_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_1 + a_1 ≤ 0 ∧ a_1 − a_1 ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  4	5	0:		a_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − ret_returnOne3_post + ret_returnOne3_post ≤ 0 ∧ ret_returnOne3_post − ret_returnOne3_post ≤ 0 ∧ − ret_returnOne3_0 + ret_returnOne3_0 ≤ 0 ∧ ret_returnOne3_0 − ret_returnOne3_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_1 + a_1 ≤ 0 ∧ a_1 − a_1 ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  4	6	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ −1 + tmp_post ≤ 0 ∧ 1 − tmp_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − ret_returnOne3_post + ret_returnOne3_post ≤ 0 ∧ ret_returnOne3_post − ret_returnOne3_post ≤ 0 ∧ − ret_returnOne3_0 + ret_returnOne3_0 ≤ 0 ∧ ret_returnOne3_0 − ret_returnOne3_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_1 + a_1 ≤ 0 ∧ a_1 − a_1 ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  1	7	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ tmp_post ≤ 0 ∧ − tmp_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − ret_returnOne3_post + ret_returnOne3_post ≤ 0 ∧ ret_returnOne3_post − ret_returnOne3_post ≤ 0 ∧ − ret_returnOne3_0 + ret_returnOne3_0 ≤ 0 ∧ ret_returnOne3_0 − ret_returnOne3_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_1 + a_1 ≤ 0 ∧ a_1 − a_1 ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  5	8	4:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + a_1 ≤ 0 ∧ −1 − a_1 ≤ 0 ∧ −1 + ret_returnOne3_post ≤ 0 ∧ 1 − ret_returnOne3_post ≤ 0 ∧ a_post − ret_returnOne3_post ≤ 0 ∧ − a_post + ret_returnOne3_post ≤ 0 ∧ a_0 − a_post ≤ 0 ∧ − a_0 + a_post ≤ 0 ∧ ret_returnOne3_0 − ret_returnOne3_post ≤ 0 ∧ − ret_returnOne3_0 + ret_returnOne3_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0
  6	9	5:		− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − ret_returnOne3_post + ret_returnOne3_post ≤ 0 ∧ ret_returnOne3_post − ret_returnOne3_post ≤ 0 ∧ − ret_returnOne3_0 + ret_returnOne3_0 ≤ 0 ∧ ret_returnOne3_0 − ret_returnOne3_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_1 + a_1 ≤ 0 ∧ a_1 − a_1 ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_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)		−1 + a_post ≤ 0 ∧ −1 + ret_returnOne3_post ≤ 0 ∧ 1 − ret_returnOne3_post ≤ 0 ∧ 1 + a_1 ≤ 0 ∧ −1 − a_1 ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ −1 + ret_returnOne3_0 ≤ 0 ∧ 1 − ret_returnOne3_0 ≤ 0
  1		(1)		−1 + a_post ≤ 0 ∧ −1 + ret_returnOne3_post ≤ 0 ∧ 1 − ret_returnOne3_post ≤ 0 ∧ 1 + a_1 ≤ 0 ∧ −1 − a_1 ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ −1 + ret_returnOne3_0 ≤ 0 ∧ 1 − ret_returnOne3_0 ≤ 0
  2		(2)		−1 + a_post ≤ 0 ∧ −1 + ret_returnOne3_post ≤ 0 ∧ 1 − ret_returnOne3_post ≤ 0 ∧ 1 + a_1 ≤ 0 ∧ −1 − a_1 ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ −1 + ret_returnOne3_0 ≤ 0 ∧ 1 − ret_returnOne3_0 ≤ 0 ∧ −1 + tmp_post ≤ 0 ∧ −1 + tmp_0 ≤ 0
  3		(3)		−1 + a_post ≤ 0 ∧ −1 + ret_returnOne3_post ≤ 0 ∧ 1 − ret_returnOne3_post ≤ 0 ∧ 1 + a_1 ≤ 0 ∧ −1 − a_1 ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ −1 + ret_returnOne3_0 ≤ 0 ∧ 1 − ret_returnOne3_0 ≤ 0 ∧ −1 + tmp_post ≤ 0 ∧ −1 + tmp_0 ≤ 0
  4		(4)		−1 + a_post ≤ 0 ∧ −1 + ret_returnOne3_post ≤ 0 ∧ 1 − ret_returnOne3_post ≤ 0 ∧ 1 + a_1 ≤ 0 ∧ −1 − a_1 ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ −1 + ret_returnOne3_0 ≤ 0 ∧ 1 − ret_returnOne3_0 ≤ 0
  5		(5)		TRUE
  6		(6)		TRUE

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

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

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 using the following ranking functions, which are bounded by −16.

4 SCC Decomposition

There exist no SCC in the program graph.

Tool configuration

T2Cert

• version: 1.0