LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 4

Transitions: (pre-variables and post-variables)

0	0	1:	− z_0 + z_0 ≤ 0 ∧ z_0 − z_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
2	1	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − y_0 + z_0 ≤ 0 ∧ −1 − x_0 + x_post ≤ 0 ∧ 1 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0
2	2	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + y_0 − z_0 ≤ 0 ∧ −1 − y_0 + y_post ≤ 0 ∧ 1 + y_0 − y_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
3	3	2:	− z_0 + z_0 ≤ 0 ∧ z_0 − z_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
1	4	2:	1 + x_0 − y_0 ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_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
4	5	0:	− z_0 + z_0 ≤ 0 ∧ z_0 − z_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

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

1	6	1:	− z_0 + z_0 ≤ 0 ∧ z_0 − z_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
2	13	2:	− z_0 + z_0 ≤ 0 ∧ z_0 − z_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

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

2 Transition Removal

We remove transitions 0, 5 using the following ranking functions, which are bounded by −13.

4:	0
0:	0
1:	0
2:	0
3:	0
4:	−4
0:	−5
1:	−6
2:	−6
3:	−6
1_var_snapshot:	−6
1*:	−6
2_var_snapshot:	−6
2*:	−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] ]
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] ]
4	lexWeak[ [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] ]
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.

1* 9 1: − z_0 + z_0 ≤ 0 ∧ z_0 − z_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

4 Location Addition

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

1 7 1_var_snapshot: − z_0 + z_0 ≤ 0 ∧ z_0 − z_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

5 Location Addition

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

2* 16 2: − z_0 + z_0 ≤ 0 ∧ z_0 − z_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

6 Location Addition

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

2 14 2_var_snapshot: − z_0 + z_0 ≤ 0 ∧ z_0 − z_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

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 { 1, 2, 3, 1_var_snapshot, 1*, 2_var_snapshot, 2* }.

7.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 0.

1:	− y_0 + z_0
2:	− y_0 + z_0
3:	− y_0 + z_0
1_var_snapshot:	− y_0 + z_0
1*:	− y_0 + z_0
2_var_snapshot:	− y_0 + z_0
2*:	− y_0 + z_0

Hints:

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

7.1.2 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 7.

1:	1 − 8⋅x_0 + 8⋅y_0
2:	−4 − 8⋅x_0 + 8⋅y_0
3:	−2 − 8⋅x_0 + 8⋅y_0
1_var_snapshot:	−8⋅x_0 + 8⋅y_0
1*:	2 − 8⋅x_0 + 8⋅y_0
2_var_snapshot:	−5 − 8⋅x_0 + 8⋅y_0
2*:	−3 − 8⋅x_0 + 8⋅y_0

Hints:

7	lexWeak[ [0, 0, 0, 0, 8, 0, 0, 0, 0, 8] ]
9	lexWeak[ [0, 0, 0, 0, 8, 0, 0, 0, 0, 8] ]
14	lexWeak[ [0, 0, 0, 0, 8, 0, 0, 0, 0, 8] ]
16	lexWeak[ [0, 0, 0, 0, 8, 0, 0, 0, 0, 8] ]
1	lexWeak[ [0, 0, 0, 0, 8, 0, 8, 0, 0, 0, 0, 8, 0] ]
3	lexWeak[ [0, 0, 0, 0, 8, 0, 0, 0, 0, 8] ]
4	lexStrict[ [0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 8] , [8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.1.3 Transition Removal

We remove transitions 9, 14, 16, 1, 3 using the following ranking functions, which are bounded by −6.

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

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] ]
1	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] ]
3	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.

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