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 ∧ i_0 − j_0 ≤ 0 ∧ −1 − i_0 + i_post ≤ 0 ∧ 1 + i_0 − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
0	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − i_0 + j_0 ≤ 0 ∧ −1 − j_0 + j_post ≤ 0 ∧ 1 + j_0 − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
2	2	0:	− n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
1	3	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + i_0 − n_0 ≤ 0 ∧ j_post ≤ 0 ∧ − j_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
3	4	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0
4	5	3:	− n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

0	6	0:	− n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0
1	13	1:	− n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

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

2 Transition Removal

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

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

Hints:

7	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
14	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0	lexWeak[ [0, 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] ]
3	lexWeak[ [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] ]
5	lexStrict[ [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* 9 0: − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

4 Location Addition

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

0 7 0_var_snapshot: − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

5 Location Addition

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

1* 16 1: − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

6 Location Addition

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

1 14 1_var_snapshot: − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0

7 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

7.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 2, 1, 0_var_snapshot, 0*, 1_var_snapshot, 1* }.

7.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 6.

0:	−4 − 7⋅i_0 + 7⋅n_0
2:	−4 − 7⋅i_0 + 7⋅n_0
1:	−7⋅i_0 + 7⋅n_0
0_var_snapshot:	−4 − 7⋅i_0 + 7⋅n_0
0*:	−4 − 7⋅i_0 + 7⋅n_0
1_var_snapshot:	−7⋅i_0 + 7⋅n_0
1*:	1 − 7⋅i_0 + 7⋅n_0

Hints:

7	lexWeak[ [7, 0, 0, 0, 0, 0, 0, 0, 0, 7] ]
9	lexWeak[ [7, 0, 0, 0, 0, 0, 0, 0, 0, 7] ]
14	lexWeak[ [7, 0, 0, 0, 0, 0, 0, 0, 0, 7] ]
16	lexWeak[ [7, 0, 0, 0, 0, 0, 0, 0, 0, 7] ]
0	lexWeak[ [0, 0, 0, 0, 7, 0, 7, 7, 0, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 7] ]
2	lexWeak[ [7, 0, 0, 0, 0, 0, 0, 0, 0, 7] ]
3	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 7] , [0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.1.2 Transition Removal

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

0:	−1 + 3⋅i_0 − 3⋅j_0
2:	1 + 3⋅i_0 − 3⋅j_0
1:	−6 + i_0 + 2⋅i_post − 3⋅j_0
0_var_snapshot:	−1 + 3⋅i_0 − 3⋅j_0
0*:	3⋅i_0 − 3⋅j_0
1_var_snapshot:	−7 + i_0 + 2⋅i_post − 3⋅j_0
1*:	−5 + i_0 + 2⋅i_post − 3⋅j_0

Hints:

7	lexWeak[ [0, 0, 0, 0, 0, 3, 0, 0, 3, 0] ]
9	lexWeak[ [0, 0, 0, 0, 0, 3, 0, 0, 3, 0] ]
14	lexWeak[ [0, 0, 0, 0, 0, 3, 2, 0, 1, 0] ]
16	lexWeak[ [0, 0, 0, 0, 0, 3, 2, 0, 1, 0] ]
0	lexWeak[ [0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 3] ]
1	lexStrict[ [0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 3, 0] , [0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 3, 0, 0, 3, 0] ]

7.1.3 Transition Removal

We remove transitions 9, 14, 16, 0, 2 using the following ranking functions, which are bounded by −6.

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

Hints:

7	lexWeak[ [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] ]
14	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
16	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 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] ]
2	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.1.4 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −1.

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

Hints:

7	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.1.5 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

7.1.5.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 6.

7.1.5.1.1 Splitting Cut-Point Transitions

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

7.1.5.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 13.

7.1.5.2.1 Splitting Cut-Point Transitions

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

Tool configuration

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