LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 4 0: − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 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 2, 3 using the following ranking functions, which are bounded by −11.

3: 0
2: 0
0: 0
1: 0
3: −4
2: −5
0: −6
1: −6
0_var_snapshot: −6
0*: −6
Hints:
5 lexWeak[ [0, 0, 0, 0] ]
0 lexWeak[ [1, 1, 0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0] ]
2 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
3 lexStrict[ [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* 7 0: − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 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 5 0_var_snapshot: − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 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 transitions 5, 7, 0, 1 using the following ranking functions, which are bounded by −4.

0: −2
1: 0
0_var_snapshot: −3
0*: −1
Hints:
5 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
7 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]
0 lexStrict[ [2, 2, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ]
1 lexStrict[ [0, 0, 0, 0] , [0, 0, 0, 0] ]

5.1.2 Splitting Cut-Point Transitions

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

5.1.2.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 4.

5.1.2.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert