LTS Termination Proof

by T2Cert

Input

Integer Transition System
• Initial Location: 7
• Transitions: (pre-variables and post-variables)  0 0 1: 1 − ox_0 + x_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0 0 1 2: ox_0 − x_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0 3 2 1: c_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0 3 3 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ ox_post − x_0 ≤ 0 ∧ − ox_post + x_0 ≤ 0 ∧ −1 + c_post ≤ 0 ∧ 1 − c_post ≤ 0 ∧ c_0 − c_post ≤ 0 ∧ − c_0 + c_post ≤ 0 ∧ ox_0 − ox_post ≤ 0 ∧ − ox_0 + ox_post ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 4 4 3: c_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0 4 5 0: 1 − c_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0 5 6 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − y_0 + y_post ≤ 0 ∧ −1 + y_0 − y_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0 5 7 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_0 + x_post ≤ 0 ∧ −1 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0 1 8 5: 1 − x_0 ≤ 0 ∧ 1 − y_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0 6 9 1: c_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0 7 10 6: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0

Proof

The following invariants are asserted.

 0: 1 − c_0 ≤ 0 1: TRUE 2: 1 − c_0 ≤ 0 3: c_0 ≤ 0 4: TRUE 5: 1 − x_0 ≤ 0 ∧ 1 − y_0 ≤ 0 6: TRUE 7: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) 1 − c_0 ≤ 0 1 (1) TRUE 2 (2) 1 − c_0 ≤ 0 3 (3) c_0 ≤ 0 4 (4) TRUE 5 (5) 1 − x_0 ≤ 0 ∧ 1 − y_0 ≤ 0 6 (6) TRUE 7 (7) TRUE
• initial node: 7
• cover edges:
• transition edges:  0 0 1 0 1 2 1 8 5 3 2 1 3 3 1 4 4 3 4 5 0 5 6 4 5 7 4 6 9 1 7 10 6

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 11 1: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 ∧ − ox_post + ox_post ≤ 0 ∧ ox_post − ox_post ≤ 0 ∧ − ox_0 + ox_0 ≤ 0 ∧ ox_0 − ox_0 ≤ 0 ∧ − c_post + c_post ≤ 0 ∧ c_post − c_post ≤ 0 ∧ − c_0 + c_0 ≤ 0 ∧ c_0 − c_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 1, 9, 10 using the following ranking functions, which are bounded by −13.

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

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

1* 14 1: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0ox_post + ox_post ≤ 0ox_postox_post ≤ 0ox_0 + ox_0 ≤ 0ox_0ox_0 ≤ 0c_post + c_post ≤ 0c_postc_post ≤ 0c_0 + c_0 ≤ 0c_0c_0 ≤ 0

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

1 12 1_var_snapshot: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0ox_post + ox_post ≤ 0ox_postox_post ≤ 0ox_0 + ox_0 ≤ 0ox_0ox_0 ≤ 0c_post + c_post ≤ 0c_postc_post ≤ 0c_0 + c_0 ≤ 0c_0c_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, 3, 4, 5, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transitions 6, 7, 8 using the following ranking functions, which are bounded by 9.

 0: −1 + 9⋅x_0 + 8⋅y_0 1: −3 + 9⋅x_0 + 8⋅y_0 3: −1 + 9⋅x_0 + 8⋅y_0 4: 9⋅x_0 + 8⋅y_0 5: −7 + 9⋅x_0 + 8⋅y_0 1_var_snapshot: −3 + 9⋅x_0 + 8⋅y_0 1*: −2 + 9⋅x_0 + 8⋅y_0

6.1.2 Transition Removal

We remove transitions 12, 14, 0, 2, 3, 4, 5 using the following ranking functions, which are bounded by −2.

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

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 11.

6.1.3.1.1 Splitting Cut-Point Transitions

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

T2Cert

• version: 1.0