LTS Termination Proof

by T2Cert

Input

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

  0	0	1:		− tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − tmp2_post + tmp2_post ≤ 0 ∧ tmp2_post − tmp2_post ≤ 0 ∧ − tmp2_0 + tmp2_0 ≤ 0 ∧ tmp2_0 − tmp2_0 ≤ 0 ∧ − tmp1_post + tmp1_post ≤ 0 ∧ tmp1_post − tmp1_post ≤ 0 ∧ − tmp1_0 + tmp1_0 ≤ 0 ∧ tmp1_0 − tmp1_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
  2	1	0:		− tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − tmp2_post + tmp2_post ≤ 0 ∧ tmp2_post − tmp2_post ≤ 0 ∧ − tmp2_0 + tmp2_0 ≤ 0 ∧ tmp2_0 − tmp2_0 ≤ 0 ∧ − tmp1_post + tmp1_post ≤ 0 ∧ tmp1_post − tmp1_post ≤ 0 ∧ − tmp1_0 + tmp1_0 ≤ 0 ∧ tmp1_0 − tmp1_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
  2	2	0:		− tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − tmp2_post + tmp2_post ≤ 0 ∧ tmp2_post − tmp2_post ≤ 0 ∧ − tmp2_0 + tmp2_0 ≤ 0 ∧ tmp2_0 − tmp2_0 ≤ 0 ∧ − tmp1_post + tmp1_post ≤ 0 ∧ tmp1_post − tmp1_post ≤ 0 ∧ − tmp1_0 + tmp1_0 ≤ 0 ∧ tmp1_0 − tmp1_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
  3	3	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ tmp1_0 − tmp1_post ≤ 0 ∧ − tmp1_0 + tmp1_post ≤ 0 ∧ tmp2_0 − tmp2_post ≤ 0 ∧ − tmp2_0 + tmp2_post ≤ 0 ∧ tmp3_0 − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_post ≤ 0
  4	4	3:		− tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − tmp2_post + tmp2_post ≤ 0 ∧ tmp2_post − tmp2_post ≤ 0 ∧ − tmp2_0 + tmp2_0 ≤ 0 ∧ tmp2_0 − tmp2_0 ≤ 0 ∧ − tmp1_post + tmp1_post ≤ 0 ∧ tmp1_post − tmp1_post ≤ 0 ∧ − tmp1_0 + tmp1_0 ≤ 0 ∧ tmp1_0 − tmp1_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_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)		i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0 ∧ −1 + j_0 ≤ 0 ∧ 1 − j_0 ≤ 0
  1		(1)		i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0 ∧ −1 + j_0 ≤ 0 ∧ 1 − j_0 ≤ 0
  2		(2)		i_post ≤ 0 ∧ − i_post ≤ 0 ∧ −1 + j_post ≤ 0 ∧ 1 − j_post ≤ 0 ∧ i_0 ≤ 0 ∧ − i_0 ≤ 0 ∧ −1 + j_0 ≤ 0 ∧ 1 − j_0 ≤ 0
  3		(3)		TRUE
  4		(4)		TRUE

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

  0	0 1
  2	1 0
  2	2 0
  3	3 2
  4	4 3

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 1, 2, 3, 4 using the following ranking functions, which are bounded by −12.

4 SCC Decomposition

There exist no SCC in the program graph.

Tool configuration

T2Cert

• version: 1.0