LTS Termination Proof

by T2Cert

Input

• Initial Location: 8
• Transitions: (pre-variables and post-variables)

  0	0	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − j_0 + x1_post ≤ 0 ∧ j_0 − x1_post ≤ 0 ∧ −1 − j_0 + y2_post ≤ 0 ∧ 1 + j_0 − y2_post ≤ 0 ∧ tmp3_0 − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_post ≤ 0 ∧ x1_0 − x1_post ≤ 0 ∧ − x1_0 + x1_post ≤ 0 ∧ y2_0 − y2_post ≤ 0 ∧ − y2_0 + y2_post ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
  0	1	1:		− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
  2	2	3:		4 − j_0 ≤ 0 ∧ − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
  2	3	0:		−3 + j_0 ≤ 0 ∧ − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
  4	4	5:		− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
  6	5	2:		− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
  3	6	4:		− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
  3	7	4:		− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
  1	8	6:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − j_0 + j_post ≤ 0 ∧ 1 + j_0 − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0
  7	9	6:		0 ≤ 0 ∧ 0 ≤ 0 ∧ j_post ≤ 0 ∧ − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0
  8	10	7:		− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_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)		4 − j_0 ≤ 0
  4		(4)		4 − j_0 ≤ 0
  5		(5)		4 − j_0 ≤ 0
  6		(6)		TRUE
  7		(7)		TRUE
  8		(8)		TRUE

• initial node: 8
• cover edges:
• transition edges:

  0	0 1
  0	1 1
  1	8 6
  2	2 3
  2	3 0
  3	6 4
  3	7 4
  4	4 5
  6	5 2
  7	9 6
  8	10 7

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 2, 4, 6, 7, 9, 10 using the following ranking functions, which are bounded by −17.

4 Location Addition

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

6* 14 6: − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0

5 Location Addition

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

6 12 6_var_snapshot: − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_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, 6, 6_var_snapshot, 6* }.

6.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −17.

6.1.2 Transition Removal

We remove transitions 12, 14, 0, 1, 5, 8 using the following ranking functions, which are bounded by −4.

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