LTS Termination Proof

by T2Cert

Input

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

  0	0	1:		36 − counter_0 ≤ 0 ∧ − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0
  0	1	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −35 + counter_0 ≤ 0 ∧ −1 − z_0 + z_post ≤ 0 ∧ 1 + z_0 − z_post ≤ 0 ∧ −1 − counter_0 + counter_post ≤ 0 ∧ 1 + counter_0 − counter_post ≤ 0 ∧ counter_0 − counter_post ≤ 0 ∧ − counter_0 + counter_post ≤ 0 ∧ z_0 − z_post ≤ 0 ∧ − z_0 + z_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0
  3	2	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −127 + y_0 ≤ 0 ∧ z_0 − z_post ≤ 0 ∧ − z_0 + z_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0
  3	3	1:		128 − y_0 ≤ 0 ∧ − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0
  2	4	0:		− z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0
  4	5	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ counter_post ≤ 0 ∧ − counter_post ≤ 0 ∧ counter_0 − counter_post ≤ 0 ∧ − counter_0 + counter_post ≤ 0 ∧ − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0
  5	6	4:		− z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_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)		−127 + y_0 ≤ 0
  1		(1)		− counter_0 ≤ 0
  2		(2)		−127 + y_0 ≤ 0
  3		(3)		counter_post ≤ 0 ∧ − counter_post ≤ 0 ∧ counter_0 ≤ 0 ∧ − counter_0 ≤ 0
  4		(4)		TRUE
  5		(5)		TRUE

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

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

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 2, 3, 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.

2* 10 2: − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0

5 Location Addition

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

2 8 2_var_snapshot: − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_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, 2_var_snapshot, 2* }.

6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −142.

6.1.2 Transition Removal

We remove transitions 8, 4 using the following ranking functions, which are bounded by −2.

6.1.3 Transition Removal

We remove transition 10 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