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 ∧ 2 − x_6_0 ≤ 0 ∧ 3 − w_5_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − x_6_post + x_6_post ≤ 0 ∧ x_6_post − x_6_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − w_5_post + w_5_post ≤ 0 ∧ w_5_post − w_5_post ≤ 0 ∧ − w_5_0 + w_5_0 ≤ 0 ∧ w_5_0 − w_5_0 ≤ 0
0	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 2 − x_6_0 ≤ 0 ∧ −2 + w_5_0 ≤ 0 ∧ −1 − x_6_0 + x_6_post ≤ 0 ∧ 1 + x_6_0 − x_6_post ≤ 0 ∧ −1 − w_5_0 + w_5_post ≤ 0 ∧ 1 + w_5_0 − w_5_post ≤ 0 ∧ w_5_0 − w_5_post ≤ 0 ∧ − w_5_0 + w_5_post ≤ 0 ∧ x_6_0 − x_6_post ≤ 0 ∧ − x_6_0 + x_6_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
2	2	0:	− x_6_post + x_6_post ≤ 0 ∧ x_6_post − x_6_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − w_5_post + w_5_post ≤ 0 ∧ w_5_post − w_5_post ≤ 0 ∧ − w_5_0 + w_5_0 ≤ 0 ∧ w_5_0 − w_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	3	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + x_6_0 ≤ 0 ∧ −1 − x_6_0 + x_6_post ≤ 0 ∧ 1 + x_6_0 − x_6_post ≤ 0 ∧ −1 − w_5_0 + w_5_post ≤ 0 ∧ 1 + w_5_0 − w_5_post ≤ 0 ∧ w_5_0 − w_5_post ≤ 0 ∧ − w_5_0 + w_5_post ≤ 0 ∧ x_6_0 − x_6_post ≤ 0 ∧ − x_6_0 + x_6_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
3	4	0:	− x_6_post + x_6_post ≤ 0 ∧ x_6_post − x_6_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − w_5_post + w_5_post ≤ 0 ∧ w_5_post − w_5_post ≤ 0 ∧ − w_5_0 + w_5_0 ≤ 0 ∧ w_5_0 − w_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	5	0:	− x_6_post + x_6_post ≤ 0 ∧ x_6_post − x_6_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − w_5_post + w_5_post ≤ 0 ∧ w_5_post − w_5_post ≤ 0 ∧ − w_5_0 + w_5_0 ≤ 0 ∧ w_5_0 − w_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	6	4:	− x_6_post + x_6_post ≤ 0 ∧ x_6_post − x_6_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − w_5_post + w_5_post ≤ 0 ∧ w_5_post − w_5_post ≤ 0 ∧ − w_5_0 + w_5_0 ≤ 0 ∧ w_5_0 − w_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:

− x_6_post + x_6_post ≤ 0 ∧ x_6_post − x_6_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − w_5_post + w_5_post ≤ 0 ∧ w_5_post − w_5_post ≤ 0 ∧ − w_5_0 + w_5_0 ≤ 0 ∧ w_5_0 − w_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 0, 5, 6 using the following ranking functions, which are bounded by −13.

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

Hints:

8	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, 0, 0, 0, 0, 0] ]
2	lexWeak[ [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, 0, 0, 0, 0] ]
4	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, 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] ]
6	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* 10 0: − x_6_post + x_6_post ≤ 0 ∧ x_6_post − x_6_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − w_5_post + w_5_post ≤ 0 ∧ w_5_post − w_5_post ≤ 0 ∧ − w_5_0 + w_5_0 ≤ 0 ∧ w_5_0 − w_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: − x_6_post + x_6_post ≤ 0 ∧ x_6_post − x_6_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − w_5_post + w_5_post ≤ 0 ∧ w_5_post − w_5_post ≤ 0 ∧ − w_5_0 + w_5_0 ≤ 0 ∧ w_5_0 − w_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, 2, 3, 0_var_snapshot, 0* }.

5.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −12.

0:	−2 − 4⋅w_5_0
2:	−4⋅w_5_0
3:	−4⋅w_5_0
0_var_snapshot:	−3 − 4⋅w_5_0
0*:	−1 − 4⋅w_5_0

Hints:

8	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
10	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
1	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0] ]
4	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0] ]

5.1.2 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −5.

0:	−1 − 3⋅x_6_0
2:	1 − 3⋅x_6_0
3:	1 − 3⋅x_6_0
0_var_snapshot:	−1 − 3⋅x_6_0
0*:	−3⋅x_6_0

Hints:

8	lexWeak[ [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]
10	lexWeak[ [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexStrict[ [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0] , [0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4	lexWeak[ [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]

5.1.3 Transition Removal

We remove transitions 8, 10, 2 using the following ranking functions, which are bounded by −3.

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

Hints:

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] ]
10	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] ]
2	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] ]
4	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

5.1.4 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −1.

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

Hints:

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] ]

5.1.5 Splitting Cut-Point Transitions

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

5.1.5.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

5.1.5.1.1 Splitting Cut-Point Transitions

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

Tool configuration

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