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 ∧ i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ i_14_1 ≤ 0 ∧ −9 + i_14_1 ≤ 0 ∧ −1 − i_14_1 + i_14_2 ≤ 0 ∧ 1 + i_14_1 − i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ −9 + i_14_2 ≤ 0 ∧ −1 − i_14_2 + i_14_post ≤ 0 ∧ 1 + i_14_2 − i_14_post ≤ 0 ∧ 2 − i_14_post ≤ 0 ∧ −2 + i_14_post ≤ 0 ∧ i_14_0 − i_14_post ≤ 0 ∧ − i_14_0 + i_14_post ≤ 0 ∧ − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0
  1	1	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 10 − i_14_0 ≤ 0 ∧ result_12_post − temp0_15_0 ≤ 0 ∧ − result_12_post + temp0_15_0 ≤ 0 ∧ result_12_0 − result_12_post ≤ 0 ∧ − result_12_0 + result_12_post ≤ 0 ∧ − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_0 ≤ 0
  1	2	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + i_14_0 ≤ 0 ∧ −1 − i_14_0 + i_14_post ≤ 0 ∧ 1 + i_14_0 − i_14_post ≤ 0 ∧ −1 + i_14_post − i_22_0 ≤ 0 ∧ 1 − i_14_post + i_22_0 ≤ 0 ∧ −9 + i_22_0 ≤ 0 ∧ i_14_0 − i_14_post ≤ 0 ∧ − i_14_0 + i_14_post ≤ 0 ∧ − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0
  3	3	1:		− temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_0 ≤ 0
  4	4	0:		− temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_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)		i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0
  2		(2)		i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 ∧ 10 − i_14_0 ≤ 0
  3		(3)		i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 ∧ −9 + i_22_0 ≤ 0
  4		(4)		TRUE

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

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

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 1, 4 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* 8 1: − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_0 ≤ 0

5 Location Addition

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

1 6 1_var_snapshot: − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_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 2 using the following ranking functions, which are bounded by −40.

6.1.2 Transition Removal

We remove transitions 6, 8, 3 using the following ranking functions, which are bounded by −1.

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