LTS Termination Proof

by T2Cert

Input

• Initial Location: 5
• Transitions: (pre-variables and post-variables)

  0	0	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_13_1 − s_17_0 ≤ 0 ∧ − i_13_1 + s_17_0 ≤ 0 ∧ j_15_post − s_16_0 ≤ 0 ∧ − j_15_post + s_16_0 ≤ 0 ∧ i_13_post − j_15_post ≤ 0 ∧ − i_13_post + j_15_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ j_15_0 − j_15_post ≤ 0 ∧ − j_15_0 + j_15_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0
  2	1	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 500 − i_13_0 ≤ 0 ∧ rt_11_post − st_14_0 ≤ 0 ∧ − rt_11_post + st_14_0 ≤ 0 ∧ rt_11_0 − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0
  2	2	4:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −499 + i_13_0 ≤ 0 ∧ −1 − i_13_0 + i_13_post ≤ 0 ∧ 1 + i_13_0 − i_13_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0
  4	3	2:		− st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0
  1	4	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 500 − i_13_0 ≤ 0 ∧ rt_11_post − st_14_0 ≤ 0 ∧ − rt_11_post + st_14_0 ≤ 0 ∧ rt_11_0 − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0
  1	5	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −499 + i_13_0 ≤ 0 ∧ −1 − i_13_0 + i_13_post ≤ 0 ∧ 1 + i_13_0 − i_13_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0
  5	6	0:		− st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_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)		500 − i_13_0 ≤ 0
  4		(4)		TRUE
  5		(5)		TRUE

• initial node: 5
• cover edges:
• transition edges:

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

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 1, 4, 5, 6 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.

2* 10 2: − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0

5 Location Addition

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

2 8 2_var_snapshot: − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_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 { 2, 4, 2_var_snapshot, 2* }.

6.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −2000.

6.1.2 Transition Removal

We remove transitions 8, 10 using the following ranking functions, which are bounded by −1.

6.1.3 Transition Removal

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

6.1.4 Splitting Cut-Point Transitions

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

6.1.4.1 Cut-Point Subproblem 1/1

6.1.4.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

• version: 1.0