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 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − e_0 ≤ 0 ∧ −100 + n_0 ≤ 0 ∧ −11 − n_0 + n_post ≤ 0 ∧ 11 + n_0 − n_post ≤ 0 ∧ −1 − e_0 + e_post ≤ 0 ∧ 1 + e_0 − e_post ≤ 0 ∧ e_0 − e_post ≤ 0 ∧ − e_0 + e_post ≤ 0 ∧ n_0 − n_post ≤ 0 ∧ − n_0 + n_post ≤ 0
1	1	0:	− n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0
0	2	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − e_0 ≤ 0 ∧ 101 − n_0 ≤ 0 ∧ 10 − n_0 + n_post ≤ 0 ∧ −10 + n_0 − n_post ≤ 0 ∧ 1 − e_0 + e_post ≤ 0 ∧ −1 + e_0 − e_post ≤ 0 ∧ e_0 − e_post ≤ 0 ∧ − e_0 + e_post ≤ 0 ∧ n_0 − n_post ≤ 0 ∧ − n_0 + n_post ≤ 0
2	3	0:	− n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0
3	4	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + e_post ≤ 0 ∧ 1 − e_post ≤ 0 ∧ e_0 − e_post ≤ 0 ∧ − e_0 + e_post ≤ 0 ∧ n_0 − n_post ≤ 0 ∧ − n_0 + n_post ≤ 0
4	5	3:	− n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_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 −11.

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

Hints:

7	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
0	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [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] ]
5	lexStrict[ [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_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_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_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − e_post + e_post ≤ 0 ∧ e_post − e_post ≤ 0 ∧ − e_0 + e_0 ≤ 0 ∧ e_0 − e_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, 1, 2, 0_var_snapshot, 0* }.

5.1.1 Transition Removal

We remove transition 0 using the following ranking functions, which are bounded by −629.

0:	−1 + 74⋅e_0 − 7⋅n_0
1:	74⋅e_0 − 7⋅n_0
2:	1 + 74⋅e_0 − 7⋅n_0
0_var_snapshot:	−2 + 74⋅e_0 − 7⋅n_0
0*:	74⋅e_0 − 7⋅n_0

Hints:

7	lexWeak[ [0, 0, 0, 7, 0, 0, 74, 0] ]
9	lexWeak[ [0, 0, 0, 7, 0, 0, 74, 0] ]
0	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 7, 74, 0, 74, 0, 0, 7] , [0, 0, 0, 0, 74, 7, 0, 0, 0, 0, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 7, 0, 0, 74, 0] ]
2	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 7, 74, 0, 74, 0, 0, 7] ]
3	lexWeak[ [0, 0, 0, 7, 0, 0, 74, 0] ]

5.1.2 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 9.

0:	−20 + 30⋅e_0
1:	30⋅e_0
2:	30⋅e_0
0_var_snapshot:	−20 + 30⋅e_0
0*:	−10 + 30⋅e_0

Hints:

7	lexWeak[ [0, 0, 0, 0, 0, 0, 30, 0] ]
9	lexWeak[ [0, 0, 0, 0, 0, 0, 30, 0] ]
1	lexWeak[ [0, 0, 0, 0, 0, 0, 30, 0] ]
2	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 30, 0, 0, 0] , [0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0, 30, 0] ]

5.1.3 Transition Removal

We remove transitions 7, 9, 1 using the following ranking functions, which are bounded by −3.

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

Hints:

7	lexStrict[ [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] ]
1	lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]
3	lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ]

5.1.4 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 0.

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

Hints:

3	lexStrict[ [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 6.

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

T2Cert

version: 1.0