LTS Termination Proof

by T2Cert

Input

Integer Transition System
• Initial Location: 7
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 − i1_0 + i1_post ≤ 0 ∧ 1 + i1_0 − i1_post ≤ 0 ∧ i1_0 − i1_post ≤ 0 ∧ − i1_0 + i1_post ≤ 0 2 1 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 3 2 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 3 3 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 3 4 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 4 5 5: 42 − i1_0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 4 6 3: −41 + i1_0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 1 7 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0 6 8 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ i1_post ≤ 0 ∧ − i1_post ≤ 0 ∧ i1_0 − i1_post ≤ 0 ∧ − i1_0 + i1_post ≤ 0 7 9 6: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0

Proof

The following invariants are asserted.

 0: TRUE 1: TRUE 2: TRUE 3: TRUE 4: TRUE 5: 42 − i1_0 ≤ 0 6: TRUE 7: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) TRUE 2 (2) TRUE 3 (3) TRUE 4 (4) TRUE 5 (5) 42 − i1_0 ≤ 0 6 (6) TRUE 7 (7) TRUE
• initial node: 7
• cover edges:
• transition edges:  0 0 1 1 7 4 2 1 0 3 2 2 3 3 0 3 4 2 4 5 5 4 6 3 6 8 1 7 9 6

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 10 1: − i1_post + i1_post ≤ 0 ∧ i1_post − i1_post ≤ 0 ∧ − i1_0 + i1_0 ≤ 0 ∧ i1_0 − i1_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 5, 8, 9 using the following ranking functions, which are bounded by −13.

 7: 0 6: 0 0: 0 1: 0 2: 0 3: 0 4: 0 5: 0 7: −5 6: −6 0: −7 1: −7 2: −7 3: −7 4: −7 1_var_snapshot: −7 1*: −7 5: −8

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

1* 13 1: i1_post + i1_post ≤ 0i1_posti1_post ≤ 0i1_0 + i1_0 ≤ 0i1_0i1_0 ≤ 0

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

1 11 1_var_snapshot: i1_post + i1_post ≤ 0i1_posti1_post ≤ 0i1_0 + i1_0 ≤ 0i1_0i1_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 { 0, 1, 2, 3, 4, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −207.

 0: −2 − 5⋅i1_0 1: 1 − 5⋅i1_0 2: −2 − 5⋅i1_0 3: −2 − 5⋅i1_0 4: −1 − 5⋅i1_0 1_var_snapshot: −5⋅i1_0 1*: 2 − 5⋅i1_0

6.1.2 Transition Removal

We remove transitions 11, 13, 0, 1, 2, 3, 4, 7 using the following ranking functions, which are bounded by −6.

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

6.1.3 Splitting Cut-Point Transitions

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

6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 10.

6.1.3.1.1 Splitting Cut-Point Transitions

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

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