LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 8

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − j_0 + x1_post ≤ 0 ∧ j_0 − x1_post ≤ 0 ∧ −1 − j_0 + y2_post ≤ 0 ∧ 1 + j_0 − y2_post ≤ 0 ∧ tmp3_0 − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_post ≤ 0 ∧ x1_0 − x1_post ≤ 0 ∧ − x1_0 + x1_post ≤ 0 ∧ y2_0 − y2_post ≤ 0 ∧ − y2_0 + y2_post ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
0	1	1:	− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
2	2	3:	4 − j_0 ≤ 0 ∧ − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
2	3	0:	−3 + j_0 ≤ 0 ∧ − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
4	4	5:	− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
6	5	2:	− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
3	6	4:	− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
3	7	4:	− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
1	8	6:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − j_0 + j_post ≤ 0 ∧ 1 + j_0 − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0
7	9	6:	0 ≤ 0 ∧ 0 ≤ 0 ∧ j_post ≤ 0 ∧ − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0
8	10	7:	− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	TRUE
2:	TRUE
3:	4 − j_0 ≤ 0
4:	4 − j_0 ≤ 0
5:	4 − j_0 ≤ 0
6:	TRUE
7:	TRUE
8:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	TRUE
2	(2)	TRUE
3	(3)	4 − j_0 ≤ 0
4	(4)	4 − j_0 ≤ 0
5	(5)	4 − j_0 ≤ 0
6	(6)	TRUE
7	(7)	TRUE
8	(8)	TRUE

initial node: 8
cover edges:
transition edges:

0 0 1

0 1 1

1 8 6

2 2 3

2 3 0

3 6 4

3 7 4

4 4 5

6 5 2

7 9 6

8 10 7

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0

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

3 Transition Removal

We remove transitions 2, 4, 6, 7, 9, 10 using the following ranking functions, which are bounded by −17.

8:	0
7:	0
0:	0
1:	0
2:	0
6:	0
3:	0
4:	0
5:	0
8:	−7
7:	−8
0:	−9
1:	−9
2:	−9
6:	−9
6_var_snapshot:	−9
6*:	−9
3:	−10
4:	−11
5:	−12

4 Location Addition

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

6* 14 6: − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0

5 Location Addition

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

6 12 6_var_snapshot: − y2_post + y2_post ≤ 0 ∧ y2_post − y2_post ≤ 0 ∧ − y2_0 + y2_0 ≤ 0 ∧ y2_0 − y2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − tmp3_post + tmp3_post ≤ 0 ∧ tmp3_post − tmp3_post ≤ 0 ∧ − tmp3_0 + tmp3_0 ≤ 0 ∧ tmp3_0 − tmp3_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_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, 6, 6_var_snapshot, 6* }.

6.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −17.

0:	1 − 6⋅j_0
1:	−6⋅j_0
2:	2 − 6⋅j_0
6:	4 − 6⋅j_0
6_var_snapshot:	3 − 6⋅j_0
6*:	5 − 6⋅j_0

6.1.2 Transition Removal

We remove transitions 12, 14, 0, 1, 5, 8 using the following ranking functions, which are bounded by −4.

0:	1
1:	0
2:	−4
6:	−2
6_var_snapshot:	−3
6*:	−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

Here we consider cut-point transition 11.

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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