LTS Termination Proof

by T2Cert

Input

• Initial Location: 8
• Transitions: (pre-variables and post-variables)

  0	0	1:		1 + y_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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  0	1	1:		oy_0 − y_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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  2	2	0:		1 + x_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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  2	3	0:		ox_0 − x_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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  2	4	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  4	5	3:		c_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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  4	6	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ ox_post − x_0 ≤ 0 ∧ − ox_post + x_0 ≤ 0 ∧ oy_post − y_0 ≤ 0 ∧ − oy_post + y_0 ≤ 0 ∧ −1 + c_post ≤ 0 ∧ 1 − c_post ≤ 0 ∧ c_0 − c_post ≤ 0 ∧ − c_0 + c_post ≤ 0 ∧ ox_0 − ox_post ≤ 0 ∧ − ox_0 + ox_post ≤ 0 ∧ oy_0 − oy_post ≤ 0 ∧ − oy_0 + oy_post ≤ 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	7	4:		c_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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  5	8	2:		1 − c_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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  6	9	5:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − y_0 + y_post ≤ 0 ∧ −1 + y_0 − y_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  6	10	5:		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 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  3	11	6:		1 − x_0 ≤ 0 ∧ 1 − y_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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  7	12	3:		c_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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
  8	13	7:		− 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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_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 − c_0 ≤ 0
  1		(1)		1 − c_0 ≤ 0
  2		(2)		1 − c_0 ≤ 0
  3		(3)		TRUE
  4		(4)		c_0 ≤ 0
  5		(5)		TRUE
  6		(6)		1 − x_0 ≤ 0 ∧ 1 − y_0 ≤ 0
  7		(7)		TRUE
  8		(8)		TRUE

• initial node: 8
• cover edges:
• transition edges:

  0	0 1
  0	1 1
  2	2 0
  2	3 0
  2	4 3
  3	11 6
  4	5 3
  4	6 3
  5	7 4
  5	8 2
  6	9 5
  6	10 5
  7	12 3
  8	13 7

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 1, 2, 3, 12, 13 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.

3* 17 3: − 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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0

5 Location Addition

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

3 15 3_var_snapshot: − 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 ∧ − oy_post + oy_post ≤ 0 ∧ oy_post − oy_post ≤ 0 ∧ − oy_0 + oy_0 ≤ 0 ∧ oy_0 − oy_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_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 { 2, 3, 4, 5, 6, 3_var_snapshot, 3* }.

6.1.1 Transition Removal

We remove transitions 9, 10, 11 using the following ranking functions, which are bounded by 9.

6.1.2 Transition Removal

We remove transitions 15, 17, 4, 5, 6, 7, 8 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.

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