LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 7 0: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − ___const_31_0 + ___const_31_0 ≤ 0 ∧ ___const_31_0 − ___const_31_0 ≤ 0 ∧ − ___const_21_0 + ___const_21_0 ≤ 0 ∧ ___const_21_0 − ___const_21_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
2 14 2: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − ___const_31_0 + ___const_31_0 ≤ 0 ∧ ___const_31_0 − ___const_31_0 ≤ 0 ∧ − ___const_21_0 + ___const_21_0 ≤ 0 ∧ ___const_21_0 − ___const_21_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 2, 6 using the following ranking functions, which are bounded by −15.

5: 0
3: 0
0: 0
2: 0
4: 0
1: 0
5: −5
3: −6
0: −7
2: −7
4: −7
0_var_snapshot: −7
0*: −7
2_var_snapshot: −7
2*: −7
1: −13

3 Location Addition

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

0* 10 0: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − ___const_31_0 + ___const_31_0 ≤ 0 ∧ ___const_31_0 − ___const_31_0 ≤ 0 ∧ − ___const_21_0 + ___const_21_0 ≤ 0 ∧ ___const_21_0 − ___const_21_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

4 Location Addition

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

0 8 0_var_snapshot: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − ___const_31_0 + ___const_31_0 ≤ 0 ∧ ___const_31_0 − ___const_31_0 ≤ 0 ∧ − ___const_21_0 + ___const_21_0 ≤ 0 ∧ ___const_21_0 − ___const_21_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

5 Location Addition

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

2* 17 2: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − ___const_31_0 + ___const_31_0 ≤ 0 ∧ ___const_31_0 − ___const_31_0 ≤ 0 ∧ − ___const_21_0 + ___const_21_0 ≤ 0 ∧ ___const_21_0 − ___const_21_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

6 Location Addition

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

2 15 2_var_snapshot: − y_6_post + y_6_post ≤ 0 ∧ y_6_post − y_6_post ≤ 0 ∧ − y_6_0 + y_6_0 ≤ 0 ∧ y_6_0 − y_6_0 ≤ 0 ∧ − x_5_post + x_5_post ≤ 0 ∧ x_5_post − x_5_post ≤ 0 ∧ − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − ___const_31_0 + ___const_31_0 ≤ 0 ∧ ___const_31_0 − ___const_31_0 ≤ 0 ∧ − ___const_21_0 + ___const_21_0 ≤ 0 ∧ ___const_21_0 − ___const_21_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

7 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

7.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 2, 4, 0_var_snapshot, 0*, 2_var_snapshot, 2* }.

7.1.1 Transition Removal

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

0: 2 − 2⋅___const_21_0 + 2⋅x_5_0
2: −2⋅___const_21_0 + 2⋅x_5_0
4: −2⋅___const_21_0 + 2⋅x_5_0
0_var_snapshot: 1 − 2⋅___const_21_0 + 2⋅x_5_0
0*: 2 − 2⋅___const_21_0 + 2⋅x_5_0
2_var_snapshot: −2⋅___const_21_0 + 2⋅x_5_0
2*: −2⋅___const_21_0 + 2⋅x_5_0

7.1.2 Transition Removal

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

0: −5 − 4⋅___const_31_0 + 4⋅y_6_0
2: −2 − 4⋅___const_31_0 + 4⋅y_6_0
4: −4⋅___const_31_0 + 4⋅y_6_0
0_var_snapshot: −6 − 4⋅___const_31_0 + 4⋅y_6_0
0*: −4 − 4⋅___const_31_0 + 4⋅y_6_0
2_var_snapshot: −3 − 4⋅___const_31_0 + 4⋅y_6_0
2*: −1 − 4⋅___const_31_0 + 4⋅y_6_0

7.1.3 Transition Removal

We remove transitions 8, 10, 15, 17, 3, 5 using the following ranking functions, which are bounded by −4.

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

7.1.4 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

7.1.4.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 7.

7.1.4.1.1 Splitting Cut-Point Transitions

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

7.1.4.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 14.

7.1.4.2.1 Splitting Cut-Point Transitions

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

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