LTS Termination Proof

by T2Cert

Input

Integer Transition System

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.

0:	−1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ a_0 ≤ 0
1:	−1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ a_0 ≤ 0
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:	−1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ −1 + b_post ≤ 0 ∧ b_0 ≤ 0
4:	−1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0
5:	TRUE
6:	TRUE

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

We consider the following cutpoint-transitions:

− 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

and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 2, 3, 4, 5, 6, 7 using the following ranking functions, which are bounded by −17.

6:	0
5:	0
4:	0
0:	0
1:	0
2:	0
3:	0
6:	−7
5:	−8
4:	−9
0:	−10
1:	−10
0_var_snapshot:	−10
0*:	−10
2:	−14
3:	−15

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.

0:	−2
1:	0
0_var_snapshot:	0
0*:	−1

6.1.2 Transition Removal

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

0:	0
1:	0
0_var_snapshot:	0
0*:	0

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

Here we consider cut-point transition 8.

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

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