LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 7

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_0 ≤ 0 ∧ Result_0 − Result_post ≤ 0 ∧ − Result_0 + Result_post ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
0	1	2:	1 + x_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0
2	2	3:	2 + y_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0
2	3	3:	− y_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0
3	4	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − x_0 + x_post ≤ 0 ∧ 1 + x_0 − x_post ≤ 0 ∧ −1 − y_0 + y_post ≤ 0 ∧ 1 + y_0 − y_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0
4	5	0:	− y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0
0	6	5:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + x_0 ≤ 0 ∧ 1 + y_0 ≤ 0 ∧ −1 − y_0 ≤ 0 ∧ −1 − y_0 + y_post ≤ 0 ∧ 1 + y_0 − y_post ≤ 0 ∧ 99 − x_0 + x_post ≤ 0 ∧ −99 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0
5	7	0:	− y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0
6	8	0:	− y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0
7	9	6:	− y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0

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

2 Transition Removal

We remove transitions 0, 8, 9 using the following ranking functions, which are bounded by −13.

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

Hints:

11	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
6	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
7	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
8	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
9	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

3 Location Addition

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

0* 13 0: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_0 ≤ 0

4 Location Addition

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

0 11 0_var_snapshot: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − Result_post + Result_post ≤ 0 ∧ Result_post − Result_post ≤ 0 ∧ − Result_0 + Result_0 ≤ 0 ∧ Result_0 − Result_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, 2, 3, 4, 5, 0_var_snapshot, 0* }.

5.1.1 Transition Removal

We remove transitions 2, 6 using the following ranking functions, which are bounded by 2.

0:	−2 − 6⋅y_0
2:	−4 − 6⋅y_0
3:	−5 − 6⋅y_0
4:	−6⋅y_0
5:	−6⋅y_0
0_var_snapshot:	−3 − 6⋅y_0
0*:	−1 − 6⋅y_0

Hints:

11	lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
13	lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexStrict[ [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] , [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0] ]
5	lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
6	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
7	lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]

5.1.2 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 2.

0:	−2 − 5⋅x_0
2:	−3 − 5⋅x_0
3:	−4 − 5⋅x_0
4:	−5⋅x_0
5:	−5⋅x_0
0_var_snapshot:	−2 − 5⋅x_0
0*:	−1 − 5⋅x_0

Hints:

11	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]
13	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]
1	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] , [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]
4	lexWeak[ [0, 0, 0, 0, 0, 5, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0] ]
5	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]
7	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]

5.1.3 Transition Removal

We remove transitions 11, 13, 3, 4, 5, 7 using the following ranking functions, which are bounded by −3.

0:	−2
2:	2
3:	1
4:	0
5:	0
0_var_snapshot:	−3
0*:	−1

Hints:

11	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
13	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
7	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

5.1.4 Splitting Cut-Point Transitions

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

5.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 10.

5.1.4.1.1 Splitting Cut-Point Transitions

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

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