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 ∧ 101 − i_5_0 ≤ 0 ∧ ____retres3_7_post ≤ 0 ∧ − ____retres3_7_post ≤ 0 ∧ ____cil_tmp4_8_post − ____retres3_7_post ≤ 0 ∧ − ____cil_tmp4_8_post + ____retres3_7_post ≤ 0 ∧ Result_4_post − ____cil_tmp4_8_post ≤ 0 ∧ − Result_4_post + ____cil_tmp4_8_post ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_post ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_0 ≤ 0
  0	1	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −100 + i_5_0 ≤ 0 ∧ 1 − x_6_0 ≤ 0 ∧ −1 − i_5_0 + i_5_post ≤ 0 ∧ 1 + i_5_0 − i_5_post ≤ 0 ∧ i_5_0 − i_5_post ≤ 0 ∧ − i_5_0 + i_5_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − ____retres3_7_post + ____retres3_7_post ≤ 0 ∧ ____retres3_7_post − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_0 ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_0 ≤ 0 ∧ − ____cil_tmp4_8_post + ____cil_tmp4_8_post ≤ 0 ∧ ____cil_tmp4_8_post − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_0 ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
  2	2	0:		− x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_0 ≤ 0 ∧ − ____retres3_7_post + ____retres3_7_post ≤ 0 ∧ ____retres3_7_post − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_0 ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_0 ≤ 0 ∧ − ____cil_tmp4_8_post + ____cil_tmp4_8_post ≤ 0 ∧ ____cil_tmp4_8_post − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_0 ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
  3	3	4:		0 ≤ 0 ∧ 0 ≤ 0 ∧ i_5_post ≤ 0 ∧ − i_5_post ≤ 0 ∧ i_5_0 − i_5_post ≤ 0 ∧ − i_5_0 + i_5_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − ____retres3_7_post + ____retres3_7_post ≤ 0 ∧ ____retres3_7_post − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_0 ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_0 ≤ 0 ∧ − ____cil_tmp4_8_post + ____cil_tmp4_8_post ≤ 0 ∧ ____cil_tmp4_8_post − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_0 ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
  4	4	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −100 + i_5_0 ≤ 0 ∧ x_6_0 ≤ 0 ∧ ____retres3_7_post ≤ 0 ∧ − ____retres3_7_post ≤ 0 ∧ ____cil_tmp4_8_post − ____retres3_7_post ≤ 0 ∧ − ____cil_tmp4_8_post + ____retres3_7_post ≤ 0 ∧ Result_4_post − ____cil_tmp4_8_post ≤ 0 ∧ − Result_4_post + ____cil_tmp4_8_post ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_post ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_0 ≤ 0
  4	5	0:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −100 + i_5_0 ≤ 0 ∧ 1 − x_6_0 ≤ 0 ∧ −1 − i_5_0 + i_5_post ≤ 0 ∧ 1 + i_5_0 − i_5_post ≤ 0 ∧ i_5_0 − i_5_post ≤ 0 ∧ − i_5_0 + i_5_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − ____retres3_7_post + ____retres3_7_post ≤ 0 ∧ ____retres3_7_post − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_0 ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_0 ≤ 0 ∧ − ____cil_tmp4_8_post + ____cil_tmp4_8_post ≤ 0 ∧ ____cil_tmp4_8_post − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_0 ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
  5	6	3:		− x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_0 ≤ 0 ∧ − ____retres3_7_post + ____retres3_7_post ≤ 0 ∧ ____retres3_7_post − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_0 ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_0 ≤ 0 ∧ − ____cil_tmp4_8_post + ____cil_tmp4_8_post ≤ 0 ∧ ____cil_tmp4_8_post − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_0 ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_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 − x_6_0 ≤ 0
  1		(1)		Result_4_post ≤ 0 ∧ ____cil_tmp4_8_post ≤ 0 ∧ ____retres3_7_post ≤ 0 ∧ − ____retres3_7_post ≤ 0 ∧ Result_4_0 ≤ 0 ∧ ____cil_tmp4_8_0 ≤ 0 ∧ ____retres3_7_0 ≤ 0 ∧ − ____retres3_7_0 ≤ 0
  2		(2)		1 − x_6_0 ≤ 0
  3		(3)		TRUE
  4		(4)		i_5_post ≤ 0 ∧ − i_5_post ≤ 0 ∧ i_5_0 ≤ 0 ∧ − i_5_0 ≤ 0
  5		(5)		TRUE

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

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

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 3, 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.

0* 10 0: − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_0 ≤ 0 ∧ − ____retres3_7_post + ____retres3_7_post ≤ 0 ∧ ____retres3_7_post − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_0 ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_0 ≤ 0 ∧ − ____cil_tmp4_8_post + ____cil_tmp4_8_post ≤ 0 ∧ ____cil_tmp4_8_post − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_0 ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

5 Location Addition

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

0 8 0_var_snapshot: − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_0 ≤ 0 ∧ − ____retres3_7_post + ____retres3_7_post ≤ 0 ∧ ____retres3_7_post − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_0 ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_0 ≤ 0 ∧ − ____cil_tmp4_8_post + ____cil_tmp4_8_post ≤ 0 ∧ ____cil_tmp4_8_post − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_0 ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_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 { 0, 2, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −404.

6.1.2 Transition Removal

We remove transitions 8, 2 using the following ranking functions, which are bounded by −1.

6.1.3 Transition Removal

We remove transition 10 using the following ranking functions, which are bounded by 0.

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