LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 4

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − x_5_0 + x_5_post ≤ 0 ∧ 1 + x_5_0 − x_5_post ≤ 0 ∧ −1 − y_6_0 + y_6_post ≤ 0 ∧ 1 + y_6_0 − y_6_post ≤ 0 ∧ 1 − z_7_0 + z_7_post ≤ 0 ∧ −1 + z_7_0 − z_7_post ≤ 0 ∧ x_5_0 − x_5_post ≤ 0 ∧ − x_5_0 + x_5_post ≤ 0 ∧ y_6_0 − y_6_post ≤ 0 ∧ − y_6_0 + y_6_post ≤ 0 ∧ z_7_0 − z_7_post ≤ 0 ∧ − z_7_0 + z_7_post ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
1	1	0:	− z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
2	2	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_5_0 + y_6_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0
2	3	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ − y_6_0 + z_7_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0
2	4	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ 1 + y_6_0 − z_7_0 ≤ 0 ∧ 1 + x_5_0 − z_7_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0
2	5	0:	1 + x_5_0 − y_6_0 ≤ 0 ∧ 1 + y_6_0 − z_7_0 ≤ 0 ∧ − x_5_0 + z_7_0 ≤ 0 ∧ − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
4	6	2:	− z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

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

2 Transition Removal

We remove transitions 2, 3, 4, 5, 6 using the following ranking functions, which are bounded by −13.

4:	0
2:	0
3:	0
0:	0
1:	0
4:	−5
2:	−6
3:	−7
0:	−8
1:	−8
0_var_snapshot:	−8
0*:	−8

3 Location Addition

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

0* 10 0: − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

4 Location Addition

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

0 8 0_var_snapshot: − z_7_post + z_7_post ≤ 0 ∧ z_7_post − z_7_post ≤ 0 ∧ − z_7_0 + z_7_0 ≤ 0 ∧ z_7_0 − z_7_0 ≤ 0 ∧ − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

5 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

5.1 SCC Subproblem 1/1

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

5.1.1 Splitting Cut-Point Transitions

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

5.1.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

5.1.1.1.1 Invariant Updates

The following invariants are asserted.

0:	1 ≤ 0
1:	1 ≤ 0
2:	TRUE
3:	TRUE
4:	TRUE
0:	1 ≤ 0
1:	1 ≤ 0
0_var_snapshot:	1 ≤ 0
0*:	1 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(4)	TRUE
1	(2)	TRUE
2	(3)	TRUE
3	(3)	TRUE
4	(3)	TRUE
5	(0)	1 ≤ 0

initial node: 0
cover edges:

2 → 4

3 → 4
transition edges:

0 6 1

1 2 2

1 3 3

1 4 4

1 5 5

5.1.1.1.2 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by −6.

0:	−1
1:	−2
0_var_snapshot:	−3
0*:	−4

5.1.1.1.3 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

version: 1.0

2	→ 4
3	→ 4