LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0
2: i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0
3: i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0
4: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 5 1: − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 1, 4 using the following ranking functions, which are bounded by −13.

4: 0
0: 0
1: 0
3: 0
2: 0
4: −5
0: −6
1: −7
3: −7
1_var_snapshot: −7
1*: −7
2: −11

4 Location Addition

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

1* 8 1: − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0

5 Location Addition

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

1 6 1_var_snapshot: − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0

6 SCC Decomposition

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

6.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 3, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −4.

1: −1 − 4⋅i_14_0 − i_14_2 + 4⋅i_22_0
3: −4⋅i_14_0 + 4⋅i_22_0
1_var_snapshot: −3 − 4⋅i_14_0 + 4⋅i_22_0
1*: −4⋅i_14_0 − i_14_2 + 4⋅i_22_0

6.1.2 Transition Removal

We remove transitions 8, 3 using the following ranking functions, which are bounded by 0.

1: 0
3: 2
1_var_snapshot: 0
1*: 1

6.1.3 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 0.

1: 1
3: 0
1_var_snapshot: 0
1*: 0

6.1.4 Splitting Cut-Point Transitions

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

6.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

6.1.4.1.1 Splitting Cut-Point Transitions

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

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