LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 2

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_0 ≤ 0 ∧ 1 − x_0 + x_post ≤ 0 ∧ −1 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0
1	1	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
2	2	0:	0 ≤ 0 ∧ 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:

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

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

Hints:

4	lexWeak[ [0, 0, 0, 0] ]
0	lexWeak[ [0, 0, 0, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 0, 0] ]
2	lexStrict[ [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* 6 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.

0 4 0_var_snapshot: − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_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, 0_var_snapshot, 0* }.

5.1.1 Transition Removal

We remove transition 0 using the following ranking functions, which are bounded by 1.

0:	−1 + 4⋅x_0
1:	1 + 4⋅x_0
0_var_snapshot:	−2 + 4⋅x_0
0*:	4⋅x_0

Hints:

4	lexWeak[ [0, 0, 4, 0] ]
6	lexWeak[ [0, 0, 4, 0] ]
0	lexStrict[ [0, 0, 0, 4, 0, 4, 0] , [0, 0, 4, 0, 0, 0, 0] ]
1	lexWeak[ [0, 0, 0, 0, 4, 0] ]

5.1.2 Transition Removal

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

0:	0
1:	2
0_var_snapshot:	−1
0*:	1

Hints:

4	lexWeak[ [0, 0, 0, 0] ]
6	lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
1	lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]

5.1.3 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 0.

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

Hints:

4	lexStrict[ [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 3.

5.1.4.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

version: 1.0