LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 3 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 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

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

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

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

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

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