LTS Termination Proof

by T2Cert

Input

• Initial Location: 9
• Transitions: (pre-variables and post-variables)

  0	0	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 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0
  1	1	0:		− z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  2	2	0:		y_0 ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  2	3	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − y_0 ≤ 0 ∧ 1 − y_0 + y_post ≤ 0 ∧ −1 + y_0 − y_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  3	4	2:		− z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  4	5	2:		− x_0 + z_0 ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  4	6	5:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_0 − z_0 ≤ 0 ∧ −1 − x_0 + x_post ≤ 0 ∧ 1 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0
  5	7	4:		− z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  6	8	4:		y_0 − z_0 ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  6	9	7:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − y_0 + z_0 ≤ 0 ∧ 1 − y_0 + y_post ≤ 0 ∧ −1 + y_0 − y_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  7	10	6:		− z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
  8	11	6:		0 ≤ 0 ∧ 0 ≤ 0 ∧ x_post ≤ 0 ∧ − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0
  9	12	8:		− z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 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)		y_0 ≤ 0
  1		(1)		y_0 ≤ 0
  2		(2)		TRUE
  3		(3)		TRUE
  4		(4)		TRUE
  5		(5)		TRUE
  6		(6)		x_post ≤ 0 ∧ − x_post ≤ 0 ∧ x_0 ≤ 0 ∧ − x_0 ≤ 0
  7		(7)		x_post ≤ 0 ∧ − x_post ≤ 0 ∧ x_0 ≤ 0 ∧ − x_0 ≤ 0
  8		(8)		TRUE
  9		(9)		TRUE

• initial node: 9
• cover edges:
• transition edges:

  0	0 1
  1	1 0
  2	2 0
  2	3 3
  3	4 2
  4	5 2
  4	6 5
  5	7 4
  6	8 4
  6	9 7
  7	10 6
  8	11 6
  9	12 8

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 2, 5, 8, 11, 12 using the following ranking functions, which are bounded by −23.

4 Location Addition

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

0* 16 0: − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 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.

0 14 0_var_snapshot: − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

6 Location Addition

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

2* 23 2: − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

7 Location Addition

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

2 21 2_var_snapshot: − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

8 Location Addition

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

4* 30 4: − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

9 Location Addition

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

4 28 4_var_snapshot: − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

10 Location Addition

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

6* 37 6: − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

11 Location Addition

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

6 35 6_var_snapshot: − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

12 SCC Decomposition

We consider subproblems for each of the 4 SCC(s) of the program graph.

12.1 SCC Subproblem 1/4

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

12.1.1 Transition Removal

We remove transition 0 using the following ranking functions, which are bounded by 0.

12.1.2 Transition Removal

We remove transitions 14, 1 using the following ranking functions, which are bounded by −3.

12.1.3 Transition Removal

We remove transition 16 using the following ranking functions, which are bounded by −1.

12.1.4 Splitting Cut-Point Transitions

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

12.1.4.1 Cut-Point Subproblem 1/1

12.1.4.1.1 Splitting Cut-Point Transitions

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

12.2 SCC Subproblem 2/4

Here we consider the SCC { 2, 3, 2_var_snapshot, 2* }.

12.2.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 1.

12.2.2 Transition Removal

We remove transitions 21, 23, 4 using the following ranking functions, which are bounded by −1.

12.2.3 Splitting Cut-Point Transitions

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

12.2.3.1 Cut-Point Subproblem 1/1

12.2.3.1.1 Splitting Cut-Point Transitions

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

12.3 SCC Subproblem 3/4

Here we consider the SCC { 4, 5, 4_var_snapshot, 4* }.

12.3.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 1.

12.3.2 Transition Removal

We remove transitions 28, 30, 7 using the following ranking functions, which are bounded by −3.

12.3.3 Splitting Cut-Point Transitions

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

12.3.3.1 Cut-Point Subproblem 1/1

12.3.3.1.1 Splitting Cut-Point Transitions

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

12.4 SCC Subproblem 4/4

Here we consider the SCC { 6, 7, 6_var_snapshot, 6* }.

12.4.1 Transition Removal

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

12.4.2 Transition Removal

We remove transitions 35, 37 using the following ranking functions, which are bounded by −2.

12.4.3 Transition Removal

We remove transition 10 using the following ranking functions, which are bounded by −1.

12.4.4 Splitting Cut-Point Transitions

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

12.4.4.1 Cut-Point Subproblem 1/1

12.4.4.1.1 Splitting Cut-Point Transitions

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

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