LTS Termination Proof

by T2Cert

Input

• Initial Location: 3
• Transitions: (pre-variables and post-variables)

  0	0	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2 ≤ 0 ∧ 2 − arg1P ≤ 0 ∧ −1 + x3 ≤ 0 ∧ − x3 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x15 + x15 ≤ 0 ∧ x15 − x15 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0
  0	1	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2 ≤ 0 ∧ 2 − arg1P ≤ 0 ∧ 3 − x7 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x3 + x3 ≤ 0 ∧ x3 − x3 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x15 + x15 ≤ 0 ∧ x15 − x15 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0
  1	2	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ arg1 − 2⋅x11 ≤ 0 ∧ − arg1 + 2⋅x11 ≤ 0 ∧ 1 − arg1 + x12 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x3 + x3 ≤ 0 ∧ x3 − x3 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x15 + x15 ≤ 0 ∧ x15 − x15 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
  2	3	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ arg1 − 2⋅x15 ≤ 0 ∧ − arg1 + 2⋅x15 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ − arg1 + 2⋅x15 ≤ 0 ∧ −1 + arg1 − 2⋅x15 ≤ 0 ∧ −1 − 2⋅arg1P + arg1 ≤ 0 ∧ 2⋅arg1P − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x3 + x3 ≤ 0 ∧ x3 − x3 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0
  1	4	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + arg1 − 2⋅x17 ≤ 0 ∧ 1 − arg1 + 2⋅x17 ≤ 0 ∧ 1 − arg1 + x18 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x3 + x3 ≤ 0 ∧ x3 − x3 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x15 + x15 ≤ 0 ∧ x15 − x15 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
  2	5	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + arg1 − 2⋅x21 ≤ 0 ∧ 1 − arg1 + 2⋅x21 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ − arg1 + 2⋅x21 ≤ 0 ∧ −1 + arg1 − 2⋅x21 ≤ 0 ∧ −1 − 2⋅arg1P + arg1 ≤ 0 ∧ 2⋅arg1P − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x3 + x3 ≤ 0 ∧ x3 − x3 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x15 + x15 ≤ 0 ∧ x15 − x15 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0
  3	6	0:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x3 + x3 ≤ 0 ∧ x3 − x3 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x15 + x15 ≤ 0 ∧ x15 − x15 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

The invariants are proved as follows.

IMPACT Invariant Proof

• nodes (location) invariant:

  0		(0)		TRUE
  1		(1)		TRUE
  2		(2)		− arg1 ≤ 0
  3		(3)		TRUE

• initial node: 3
• cover edges:
• transition edges:

  0	0 1
  0	1 1
  1	2 2
  1	4 2
  2	3 1
  2	5 1
  3	6 0

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 1, 6 using the following ranking functions, which are bounded by −11.

4 Location Addition

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

1* 10 1: − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x3 + x3 ≤ 0 ∧ x3 − x3 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x15 + x15 ≤ 0 ∧ x15 − x15 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

5 Location Addition

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

1 8 1_var_snapshot: − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x3 + x3 ≤ 0 ∧ x3 − x3 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x15 + x15 ≤ 0 ∧ x15 − x15 ≤ 0 ∧ − x12 + x12 ≤ 0 ∧ x12 − x12 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 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, 2, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transitions 2, 3, 4, 5 using the following ranking functions, which are bounded by −1.

6.1.2 Transition Removal

We remove transitions 8, 10 using the following ranking functions, which are bounded by −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

6.1.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

• version: 1.0