LTS Termination Proof

by T2Cert

Input

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

  0	0	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ x_0 ≤ 0 ∧ −1 + x_post ≤ 0 ∧ 1 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0
  0	1	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_0 ≤ 0 ∧ −1 − x_0 + x_post ≤ 0 ∧ 1 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0
  2	2	3:		4 − x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  2	3	0:		−3 + x_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  1	4	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  4	5	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0
  5	6	4:		0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_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)		−3 + x_0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
  1		(1)		−5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
  2		(2)		−5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
  3		(3)		4 − x_0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0
  4		(4)		TRUE
  5		(5)		TRUE

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

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

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 2, 5, 6 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.

1* 10 1: − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

5 Location Addition

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

1 8 1_var_snapshot: − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_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, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transitions 0, 1 using the following ranking functions, which are bounded by −45.

6.1.2 Transition Removal

We remove transitions 8, 10, 3, 4 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