LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 5

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 2 + x_5_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 ∧ 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 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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
1	1	2:	− 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 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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	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 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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
0	3	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − x_5_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 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 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_0 ≤ 0
4	4	0:	2 − k_7_0 ≤ 0 ∧ 2 − w_8_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 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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	5	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + k_7_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 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 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_0 ≤ 0
5	6	4:	− 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 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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 Invariant Updates

The following invariants are asserted.

0:	2 − k_7_0 ≤ 0 ∧ 2 − w_8_0 ≤ 0
1:	2 − k_7_0 ≤ 0 ∧ 2 − w_8_0 ≤ 0
2:	2 − k_7_0 ≤ 0 ∧ 2 − w_8_0 ≤ 0
3:	TRUE
4:	TRUE
5:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	2 − k_7_0 ≤ 0 ∧ 2 − w_8_0 ≤ 0
1	(1)	2 − k_7_0 ≤ 0 ∧ 2 − w_8_0 ≤ 0
2	(2)	2 − k_7_0 ≤ 0 ∧ 2 − w_8_0 ≤ 0
3	(3)	TRUE
4	(4)	TRUE
5	(5)	TRUE

initial node: 5
cover edges:
transition edges:

0 0 1

0 3 3

1 1 2

2 2 0

4 4 0

4 5 3

5 6 4

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− 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 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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.

3 Transition Removal

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

5:	0
4:	0
0:	0
1:	0
2:	0
3:	0
5:	−5
4:	−6
0:	−7
1:	−7
2:	−7
0_var_snapshot:	−7
0*:	−7
3:	−11

4 Location Addition

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

0* 10 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 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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 Location Addition

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

0 8 0_var_snapshot: − 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 ∧ − w_8_0 + w_8_0 ≤ 0 ∧ w_8_0 − w_8_0 ≤ 0 ∧ − k_7_0 + k_7_0 ≤ 0 ∧ k_7_0 − k_7_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

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, 2, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

We remove transition 0 using the following ranking functions, which are bounded by 7.

0:	−1 − 5⋅x_5_0
1:	2 − 5⋅x_5_0
2:	1 − 5⋅x_5_0
0_var_snapshot:	−2 − 5⋅x_5_0
0*:	−5⋅x_5_0

6.1.2 Transition Removal

We remove transitions 8, 10, 1, 2 using the following ranking functions, which are bounded by −1.

0:	0
1:	1 + w_8_0
2:	w_8_0
0_var_snapshot:	−1
0*:	−1 + w_8_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 7.

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

version: 1.0