LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: x_14_0 ≤ 0
3: 1 − x_20_0 ≤ 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: − x_20_0 + x_20_0 ≤ 0 ∧ x_20_0 − x_20_0 ≤ 0 ∧ − x_14_post + x_14_post ≤ 0 ∧ x_14_post − x_14_post ≤ 0 ∧ − x_14_0 + x_14_0 ≤ 0 ∧ x_14_0 − x_14_0 ≤ 0 ∧ − 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 ∧ − nondet_13_post + nondet_13_post ≤ 0 ∧ nondet_13_post − nondet_13_post ≤ 0 ∧ − nondet_13_1 + nondet_13_1 ≤ 0 ∧ nondet_13_1 − nondet_13_1 ≤ 0 ∧ − nondet_13_0 + nondet_13_0 ≤ 0 ∧ nondet_13_0 − nondet_13_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: − x_20_0 + x_20_0 ≤ 0 ∧ x_20_0 − x_20_0 ≤ 0 ∧ − x_14_post + x_14_post ≤ 0 ∧ x_14_post − x_14_post ≤ 0 ∧ − x_14_0 + x_14_0 ≤ 0 ∧ x_14_0 − x_14_0 ≤ 0 ∧ − 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 ∧ − nondet_13_post + nondet_13_post ≤ 0 ∧ nondet_13_post − nondet_13_post ≤ 0 ∧ − nondet_13_1 + nondet_13_1 ≤ 0 ∧ nondet_13_1 − nondet_13_1 ≤ 0 ∧ − nondet_13_0 + nondet_13_0 ≤ 0 ∧ nondet_13_0 − nondet_13_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: − x_20_0 + x_20_0 ≤ 0 ∧ x_20_0 − x_20_0 ≤ 0 ∧ − x_14_post + x_14_post ≤ 0 ∧ x_14_post − x_14_post ≤ 0 ∧ − x_14_0 + x_14_0 ≤ 0 ∧ x_14_0 − x_14_0 ≤ 0 ∧ − 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 ∧ − nondet_13_post + nondet_13_post ≤ 0 ∧ nondet_13_post − nondet_13_post ≤ 0 ∧ − nondet_13_1 + nondet_13_1 ≤ 0 ∧ nondet_13_1 − nondet_13_1 ≤ 0 ∧ − nondet_13_0 + nondet_13_0 ≤ 0 ∧ nondet_13_0 − nondet_13_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 1.

1: −1 + 4⋅x_14_0
3: 1 + 4⋅x_14_0
1_var_snapshot: −2 + 4⋅x_14_0
1*: 4⋅x_14_0

6.1.2 Transition Removal

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

1: −2
3: 0
1_var_snapshot: −3
1*: −1

6.1.3 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by −1.

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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