LTS Termination Proof

by T2Cert

Input

• Initial Location: 4
• Transitions: (pre-variables and post-variables)

  0	0	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_0 + x_post ≤ 0 ∧ −1 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0
  1	1	2:		1 − x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  1	2	2:		1 + x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  2	3	0:		0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  3	4	0:		1 − x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  4	5	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

2 Transition Removal

We remove transitions 4, 5 using the following ranking functions, which are bounded by −11.

3 Location Addition

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

0* 9 0: − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

4 Location Addition

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

0 7 0_var_snapshot: − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

5 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

5.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 1, 2, 0_var_snapshot, 0* }.

5.1.1 Transition Removal

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

5.1.2 Splitting Cut-Point Transitions

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

5.1.2.1 Cut-Point Subproblem 1/1

5.1.2.1.1 Invariant Updates

The following invariants are asserted.

The invariants are proved as follows.

IMPACT Invariant Proof

• nodes (location) invariant:

  0		(4)		TRUE
  1		(3)		TRUE
  2		(0)		1 − x_0 ≤ 0
  3		(1)		− x_0 ≤ 0
  4		(0)		1 − x_0 ≤ 0
  5		(0_var_snapshot)		1 − x_0 ≤ 0
  10		(2)		1 − x_0 ≤ 0
  11		(2)		1 ≤ 0
  17		(1)		− x_0 ≤ 0
  18		(2)		1 ≤ 0
  26		(0)		1 − x_0 ≤ 0

• initial node: 0
• cover edges:

  26

  → 2

• transition edges:

  0	5 1
  1	4 2
  2	0 3
  2	6 4
  3	1 10
  3	2 11
  4	7 5
  5	0 17
  10	3 26
  17	2 18

5.1.2.1.2 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −7.

5.1.2.1.3 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

• version: 1.0