LTS Termination Proof

by T2Cert

Input

• Initial Location: 4
• Transitions: (pre-variables and post-variables)

  0	0	1:		___const_50_0 − i_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − ___const_50_0 + ___const_50_0 ≤ 0 ∧ ___const_50_0 − ___const_50_0 ≤ 0
  0	1	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − ___const_50_0 + i_0 ≤ 0 ∧ −1 − i_0 + i_post ≤ 0 ∧ 1 + i_0 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − ___const_50_0 + ___const_50_0 ≤ 0 ∧ ___const_50_0 − ___const_50_0 ≤ 0
  2	2	0:		− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − ___const_50_0 + ___const_50_0 ≤ 0 ∧ ___const_50_0 − ___const_50_0 ≤ 0
  3	3	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_1 ≤ 0 ∧ − i_1 ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − ___const_50_0 + ___const_50_0 ≤ 0 ∧ ___const_50_0 − ___const_50_0 ≤ 0
  4	4	3:		− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − ___const_50_0 + ___const_50_0 ≤ 0 ∧ ___const_50_0 − ___const_50_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)		i_1 ≤ 0 ∧ − i_1 ≤ 0
  1		(1)		i_1 ≤ 0 ∧ − i_1 ≤ 0
  2		(2)		i_1 ≤ 0 ∧ − i_1 ≤ 0
  3		(3)		TRUE
  4		(4)		TRUE

• initial node: 4
• cover edges:
• transition edges:

  0	0 1
  0	1 2
  2	2 0
  3	3 2
  4	4 3

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 3, 4 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.

2* 8 2: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − ___const_50_0 + ___const_50_0 ≤ 0 ∧ ___const_50_0 − ___const_50_0 ≤ 0

5 Location Addition

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

2 6 2_var_snapshot: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_1 + i_1 ≤ 0 ∧ i_1 − i_1 ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − ___const_50_0 + ___const_50_0 ≤ 0 ∧ ___const_50_0 − ___const_50_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 2.

6.1.2 Transition Removal

We remove transitions 6, 8, 2 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.

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