LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: TRUE
3: x_13_0 ≤ 0
5: 1 − y_33_post ≤ 0 ∧ 1 − y_33_0 ≤ 0
6: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 8 1: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0
2 15 2: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 2, 7 using the following ranking functions, which are bounded by −15.

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

4 Location Addition

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

1* 11 1: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0

5 Location Addition

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

1 9 1_var_snapshot: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0

6 Location Addition

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

2* 18 2: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0

7 Location Addition

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

2 16 2_var_snapshot: − y_33_post + y_33_post ≤ 0 ∧ y_33_post − y_33_post ≤ 0 ∧ − y_33_0 + y_33_0 ≤ 0 ∧ y_33_0 − y_33_0 ≤ 0 ∧ − y_28_post + y_28_post ≤ 0 ∧ y_28_post − y_28_post ≤ 0 ∧ − y_28_0 + y_28_0 ≤ 0 ∧ y_28_0 − y_28_0 ≤ 0 ∧ − y_16_post + y_16_post ≤ 0 ∧ y_16_post − y_16_post ≤ 0 ∧ − y_16_0 + y_16_0 ≤ 0 ∧ y_16_0 − y_16_0 ≤ 0 ∧ − x_32_post + x_32_post ≤ 0 ∧ x_32_post − x_32_post ≤ 0 ∧ − x_32_0 + x_32_0 ≤ 0 ∧ x_32_0 − x_32_0 ≤ 0 ∧ − x_27_post + x_27_post ≤ 0 ∧ x_27_post − x_27_post ≤ 0 ∧ − x_27_0 + x_27_0 ≤ 0 ∧ x_27_0 − x_27_0 ≤ 0 ∧ − x_13_post + x_13_post ≤ 0 ∧ x_13_post − x_13_post ≤ 0 ∧ − x_13_0 + x_13_0 ≤ 0 ∧ x_13_0 − x_13_0 ≤ 0 ∧ − temp0_14_0 + temp0_14_0 ≤ 0 ∧ temp0_14_0 − temp0_14_0 ≤ 0 ∧ − result_11_post + result_11_post ≤ 0 ∧ result_11_post − result_11_post ≤ 0 ∧ − result_11_0 + result_11_0 ≤ 0 ∧ result_11_0 − result_11_0 ≤ 0

8 SCC Decomposition

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

8.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 2, 5, 1_var_snapshot, 1*, 2_var_snapshot, 2* }.

8.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 4.

1: −4 + 10⋅x_13_0
2: −1 + 10⋅x_13_0 − 6⋅y_16_0
5: 1 + 10⋅x_13_0 − 6⋅y_16_0
1_var_snapshot: −5 + 10⋅x_13_0
1*: −3 + 10⋅x_13_0
2_var_snapshot: −2 + 10⋅x_13_0 − 6⋅y_16_0
2*: 10⋅x_13_0 − 6⋅y_16_0

8.1.2 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 1.

1: −3 + 3⋅y_16_0
2: −1 + 3⋅y_16_0
5: 1 + 3⋅y_16_0
1_var_snapshot: −4 + 3⋅y_16_0
1*: −2 + 3⋅y_16_0
2_var_snapshot: −1 + 3⋅y_16_0
2*: 3⋅y_16_0

8.1.3 Transition Removal

We remove transitions 11, 16, 18, 4, 6 using the following ranking functions, which are bounded by −3.

1: −2
2: 1
5: 3
1_var_snapshot: −3
1*: −1
2_var_snapshot: 0
2*: 2

8.1.4 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by −1.

1: 0
2: 0
5: 0
1_var_snapshot: −1
1*: 0
2_var_snapshot: 0
2*: 0

8.1.5 Splitting Cut-Point Transitions

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

8.1.5.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 8.

8.1.5.1.1 Splitting Cut-Point Transitions

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

8.1.5.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 15.

8.1.5.2.1 Splitting Cut-Point Transitions

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

Tool configuration

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