LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
3 13 3: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − x0_post + x0_post ≤ 0 ∧ x0_post − x0_post ≤ 0 ∧ − x0_0 + x0_0 ≤ 0 ∧ x0_0 − x0_0 ≤ 0 ∧ − oldX3_post + oldX3_post ≤ 0 ∧ oldX3_post − oldX3_post ≤ 0 ∧ − oldX3_0 + oldX3_0 ≤ 0 ∧ oldX3_0 − oldX3_0 ≤ 0 ∧ − oldX2_post + oldX2_post ≤ 0 ∧ oldX2_post − oldX2_post ≤ 0 ∧ − oldX2_0 + oldX2_0 ≤ 0 ∧ oldX2_0 − oldX2_0 ≤ 0 ∧ − oldX1_post + oldX1_post ≤ 0 ∧ oldX1_post − oldX1_post ≤ 0 ∧ − oldX1_0 + oldX1_0 ≤ 0 ∧ oldX1_0 − oldX1_0 ≤ 0 ∧ − oldX0_post + oldX0_post ≤ 0 ∧ oldX0_post − oldX0_post ≤ 0 ∧ − oldX0_0 + oldX0_0 ≤ 0 ∧ oldX0_0 − oldX0_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12 using the following ranking functions, which are bounded by −17.

6: 0
5: 0
4: 0
2: 0
3: 0
0: 0
1: 0
6: −7
5: −8
4: −9
2: −10
3: −10
3_var_snapshot: −10
3*: −10
0: −14
1: −15

3 Location Addition

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

3* 16 3: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − x0_post + x0_post ≤ 0 ∧ x0_post − x0_post ≤ 0 ∧ − x0_0 + x0_0 ≤ 0 ∧ x0_0 − x0_0 ≤ 0 ∧ − oldX3_post + oldX3_post ≤ 0 ∧ oldX3_post − oldX3_post ≤ 0 ∧ − oldX3_0 + oldX3_0 ≤ 0 ∧ oldX3_0 − oldX3_0 ≤ 0 ∧ − oldX2_post + oldX2_post ≤ 0 ∧ oldX2_post − oldX2_post ≤ 0 ∧ − oldX2_0 + oldX2_0 ≤ 0 ∧ oldX2_0 − oldX2_0 ≤ 0 ∧ − oldX1_post + oldX1_post ≤ 0 ∧ oldX1_post − oldX1_post ≤ 0 ∧ − oldX1_0 + oldX1_0 ≤ 0 ∧ oldX1_0 − oldX1_0 ≤ 0 ∧ − oldX0_post + oldX0_post ≤ 0 ∧ oldX0_post − oldX0_post ≤ 0 ∧ − oldX0_0 + oldX0_0 ≤ 0 ∧ oldX0_0 − oldX0_0 ≤ 0

4 Location Addition

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

3 14 3_var_snapshot: − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − x0_post + x0_post ≤ 0 ∧ x0_post − x0_post ≤ 0 ∧ − x0_0 + x0_0 ≤ 0 ∧ x0_0 − x0_0 ≤ 0 ∧ − oldX3_post + oldX3_post ≤ 0 ∧ oldX3_post − oldX3_post ≤ 0 ∧ − oldX3_0 + oldX3_0 ≤ 0 ∧ oldX3_0 − oldX3_0 ≤ 0 ∧ − oldX2_post + oldX2_post ≤ 0 ∧ oldX2_post − oldX2_post ≤ 0 ∧ − oldX2_0 + oldX2_0 ≤ 0 ∧ oldX2_0 − oldX2_0 ≤ 0 ∧ − oldX1_post + oldX1_post ≤ 0 ∧ oldX1_post − oldX1_post ≤ 0 ∧ − oldX1_0 + oldX1_0 ≤ 0 ∧ oldX1_0 − oldX1_0 ≤ 0 ∧ − oldX0_post + oldX0_post ≤ 0 ∧ oldX0_post − oldX0_post ≤ 0 ∧ − oldX0_0 + oldX0_0 ≤ 0 ∧ oldX0_0 − oldX0_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 { 2, 3, 3_var_snapshot, 3* }.

5.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 3.

2: 10 − 8⋅x0_0 + 4⋅x1_0
3: 2 − 4⋅x0_0 + 4⋅x1_0
3_var_snapshot: −4⋅x0_0 + 4⋅x1_0
3*: 4 − 4⋅x0_0 + 4⋅x1_0

5.1.2 Transition Removal

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

2: 0
3: −2
3_var_snapshot: −3
3*: −1

5.1.3 Transition Removal

We remove transition 16 using the following ranking functions, which are bounded by −1.

2: 0
3: −1
3_var_snapshot: 0
3*: 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 13.

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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