LTS Termination Proof

by T2Cert

Input

• Initial Location: 6
• Transitions: (pre-variables and post-variables)

  0	0	1:		1 ≤ 0 ∧ − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  1	1	0:		− b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  0	2	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ b_post ≤ 0 ∧ − b_post ≤ 0 ∧ b_0 − b_post ≤ 0 ∧ − b_0 + b_post ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  2	3	3:		b_0 ≤ 0 ∧ − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  4	4	0:		a_0 ≤ 0 ∧ − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  4	5	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + b_post ≤ 0 ∧ 1 − b_post ≤ 0 ∧ b_0 − b_post ≤ 0 ∧ − b_0 + b_post ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  5	6	4:		0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ a_0 − a_post ≤ 0 ∧ − a_0 + a_post ≤ 0 ∧ − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0
  6	7	5:		− b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_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)		−1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ a_0 ≤ 0
  1		(1)		−1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ a_0 ≤ 0
  2		(2)		−1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ −1 + b_post ≤ 0 ∧ −1 + b_0 ≤ 0
  3		(3)		−1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ −1 + b_post ≤ 0 ∧ b_0 ≤ 0
  4		(4)		−1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0
  5		(5)		TRUE
  6		(6)		TRUE

• initial node: 6
• cover edges:
• transition edges:

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

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 2, 3, 4, 5, 6, 7 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.

0* 11 0: − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_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: − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_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, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

We remove transitions 11, 0, 1 using the following ranking functions, which are bounded by −3.

6.1.2 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by −1.

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