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 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ 1 − b_7_0 ≤ 0 ∧ b_7_post ≤ 0 ∧ − b_7_post ≤ 0 ∧ 1 − y_6_0 + y_6_post ≤ 0 ∧ −1 + y_6_0 − y_6_post ≤ 0 ∧ b_7_0 − b_7_post ≤ 0 ∧ − b_7_0 + b_7_post ≤ 0 ∧ y_6_0 − y_6_post ≤ 0 ∧ − y_6_0 + y_6_post ≤ 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
0	1	2:	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 ∧ − 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 ∧ − b_7_post + b_7_post ≤ 0 ∧ b_7_post − b_7_post ≤ 0 ∧ − b_7_0 + b_7_0 ≤ 0 ∧ b_7_0 − b_7_0 ≤ 0
3	2	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ b_7_post ≤ 0 ∧ − b_7_post ≤ 0 ∧ b_7_0 − b_7_post ≤ 0 ∧ − b_7_0 + b_7_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 ∧ − 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	3	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_5_0 − y_6_0 ≤ 0 ∧ b_7_0 ≤ 0 ∧ −1 + b_7_post ≤ 0 ∧ 1 − b_7_post ≤ 0 ∧ −1 − x_5_0 + x_5_post ≤ 0 ∧ 1 + x_5_0 − x_5_post ≤ 0 ∧ b_7_0 − b_7_post ≤ 0 ∧ − b_7_0 + b_7_post ≤ 0 ∧ x_5_0 − x_5_post ≤ 0 ∧ − x_5_0 + x_5_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 ∧ − 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	4	2:	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 ∧ − 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 ∧ − b_7_post + b_7_post ≤ 0 ∧ b_7_post − b_7_post ≤ 0 ∧ − b_7_0 + b_7_0 ≤ 0 ∧ b_7_0 − b_7_0 ≤ 0
4	5	3:	− 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 ∧ − b_7_post + b_7_post ≤ 0 ∧ b_7_post − b_7_post ≤ 0 ∧ − b_7_0 + b_7_0 ≤ 0 ∧ b_7_0 − b_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:	−1 + b_7_post ≤ 0 ∧ 1 − b_7_post ≤ 0 ∧ −1 + b_7_0 ≤ 0 ∧ 1 − b_7_0 ≤ 0
1:	b_7_post ≤ 0 ∧ − b_7_post ≤ 0 ∧ b_7_0 ≤ 0 ∧ − b_7_0 ≤ 0
2:	− b_7_post ≤ 0 ∧ − b_7_0 ≤ 0
3:	TRUE
4:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	−1 + b_7_post ≤ 0 ∧ 1 − b_7_post ≤ 0 ∧ −1 + b_7_0 ≤ 0 ∧ 1 − b_7_0 ≤ 0
1	(1)	b_7_post ≤ 0 ∧ − b_7_post ≤ 0 ∧ b_7_0 ≤ 0 ∧ − b_7_0 ≤ 0
2	(2)	− b_7_post ≤ 0 ∧ − b_7_0 ≤ 0
3	(3)	TRUE
4	(4)	TRUE

initial node: 4
cover edges:
transition edges:

0 0 1

0 1 2

1 3 0

1 4 2

3 2 1

4 5 3

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 ∧ − b_7_post + b_7_post ≤ 0 ∧ b_7_post − b_7_post ≤ 0 ∧ − b_7_0 + b_7_0 ≤ 0 ∧ b_7_0 − b_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 1, 2, 4, 5 using the following ranking functions, which are bounded by −13.

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

4 Location Addition

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

1* 9 1: − 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 ∧ − b_7_post + b_7_post ≤ 0 ∧ b_7_post − b_7_post ≤ 0 ∧ − b_7_0 + b_7_0 ≤ 0 ∧ b_7_0 − b_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.

1 7 1_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 ∧ − b_7_post + b_7_post ≤ 0 ∧ b_7_post − b_7_post ≤ 0 ∧ − b_7_0 + b_7_0 ≤ 0 ∧ b_7_0 − b_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, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transitions 0, 3 using the following ranking functions, which are bounded by 3.

0:	3⋅b_7_post − 4⋅x_5_0 + 4⋅y_6_0
1:	1 − 4⋅x_5_0 + 4⋅y_6_0
1_var_snapshot:	−4⋅x_5_0 + 4⋅y_6_0
1*:	2 − 4⋅x_5_0 + 4⋅y_6_0

6.1.2 Transition Removal

We remove transitions 7, 9 using the following ranking functions, which are bounded by −2.

0:	0
1:	−1
1_var_snapshot:	−2
1*:	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 6.

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