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 ∧ 2 + x_5_0 ≤ 0 ∧ −1 − x_5_0 + x_5_post ≤ 0 ∧ 1 + x_5_0 − x_5_post ≤ 0 ∧ −1 − y_6_0 + y_6_post ≤ 0 ∧ 1 + y_6_0 − y_6_post ≤ 0 ∧ x_5_0 − x_5_post ≤ 0 ∧ − x_5_0 + x_5_post ≤ 0 ∧ y_6_0 − y_6_post ≤ 0 ∧ − y_6_0 + y_6_post ≤ 0 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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
  1	1	2:		1 + y_6_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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
  1	2	2:		1 − y_6_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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	3	0:		− y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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
  0	4	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − x_5_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_0 ≤ 0
  4	5	0:		2 − k_7_0 ≤ 0 ∧ 2 − w_8_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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	6	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + k_7_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_0 ≤ 0
  5	7	4:		− y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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)		2 − k_7_0 ≤ 0 ∧ 2 − w_8_0 ≤ 0
  1		(1)		2 − k_7_0 ≤ 0 ∧ 2 − w_8_0 ≤ 0
  2		(2)		2 − k_7_0 ≤ 0 ∧ 2 − w_8_0 ≤ 0
  3		(3)		TRUE
  4		(4)		TRUE
  5		(5)		TRUE

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

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

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 4, 5, 6, 7 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.

0* 11 0: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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 9 0_var_snapshot: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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, 1, 2, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 5.

6.1.2 Transition Removal

We remove transition 0 using the following ranking functions, which are bounded by 4.

6.1.3 Transition Removal

We remove transitions 9, 11, 2, 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