LTS Termination Proof

by T2Cert

Input

Integer Transition System
• Initial Location: 4
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − i_5_0 ≤ 0 ∧ −1 − i_5_0 + i_5_1 ≤ 0 ∧ 1 + i_5_0 − i_5_1 ≤ 0 ∧ 2 − i_5_1 + i_5_post ≤ 0 ∧ −2 + i_5_1 − i_5_post ≤ 0 ∧ i_5_0 − i_5_post ≤ 0 ∧ − i_5_0 + i_5_post ≤ 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 1 1 0: − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_1 + i_5_1 ≤ 0 ∧ i_5_1 − i_5_1 ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_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 0 2 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + i_5_0 ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_1 + i_5_1 ≤ 0 ∧ i_5_1 − i_5_1 ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_0 ≤ 0 3 3 0: − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_1 + i_5_1 ≤ 0 ∧ i_5_1 − i_5_1 ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_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 4 3: − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_1 + i_5_1 ≤ 0 ∧ i_5_1 − i_5_1 ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_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

Proof

The following invariants are asserted.

 0: TRUE 1: TRUE 2: 1 + i_5_0 ≤ 0 3: TRUE 4: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) TRUE 2 (2) 1 + i_5_0 ≤ 0 3 (3) TRUE 4 (4) TRUE
• initial node: 4
• cover edges:
• transition edges:  0 0 1 0 2 2 1 1 0 3 3 0 4 4 3

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 0 5 0: − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_1 + i_5_1 ≤ 0 ∧ i_5_1 − i_5_1 ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_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.

3 Transition Removal

We remove transitions 2, 3, 4 using the following ranking functions, which are bounded by −13.

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

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

0* 8 0: i_5_post + i_5_post ≤ 0i_5_posti_5_post ≤ 0i_5_1 + i_5_1 ≤ 0i_5_1i_5_1 ≤ 0i_5_0 + i_5_0 ≤ 0i_5_0i_5_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 0

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

0 6 0_var_snapshot: i_5_post + i_5_post ≤ 0i_5_posti_5_post ≤ 0i_5_1 + i_5_1 ≤ 0i_5_1i_5_1 ≤ 0i_5_0 + i_5_0 ≤ 0i_5_0i_5_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_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, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

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

 0: −2 + 4⋅i_5_0 1: 4⋅i_5_0 0_var_snapshot: −3 + 4⋅i_5_0 0*: −1 + 4⋅i_5_0

6.1.2 Transition Removal

We remove transitions 6, 8, 1 using the following ranking functions, which are bounded by −3.

 0: −2 1: 0 0_var_snapshot: −3 0*: −1

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 5.

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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