LTS Termination Proof

by T2Cert

Input

• Initial Location: 2
• Transitions: (pre-variables and post-variables)

  0	0	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0
  1	1	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ −1 − arg1P + arg2 ≤ 0 ∧ 1 + arg1P − arg2 ≤ 0 ∧ arg1 − arg2P ≤ 0 ∧ − arg1 + arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0
  2	2	0:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

2 Transition Removal

We remove transitions 0, 2 using the following ranking functions, which are bounded by −11.

3 Location Addition

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

1* 6 1: − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

4 Location Addition

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

1 4 1_var_snapshot: − 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 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

5.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 1_var_snapshot, 1* }.

5.1.1 Splitting Cut-Point Transitions

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

5.1.1.1 Cut-Point Subproblem 1/1

5.1.1.1.1 Fresh Variable Addition

The new variable __snapshot_1_arg2P is introduced. The transition formulas are extended as follows:

5.1.1.1.2 Fresh Variable Addition

The new variable __snapshot_1_arg2 is introduced. The transition formulas are extended as follows:

5.1.1.1.3 Fresh Variable Addition

The new variable __snapshot_1_arg1P is introduced. The transition formulas are extended as follows:

5.1.1.1.4 Fresh Variable Addition

The new variable __snapshot_1_arg1 is introduced. The transition formulas are extended as follows:

5.1.1.1.5 Invariant Updates

The following invariants are asserted.

The invariants are proved as follows.

IMPACT Invariant Proof

• nodes (location) invariant:

  0		(2)		TRUE
  1		(0)		TRUE
  2		(1)		arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ − arg1P − arg2P ≤ 0 ∧ − arg2 ≤ 0
  3		(1)		arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0
  4		(1)		arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ − arg1P − arg2P ≤ 0 ∧ − arg2 ≤ 0
  5		(1_var_snapshot)		− __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0 ∧ − arg2 ≤ 0
  12		(1*)		arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0
  13		(1)		arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0
  14		(1_var_snapshot)		− __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0
  15		(1_var_snapshot)		− __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0
  23		(1)		arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0
  24		(1)		arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0
  28		(1*)		arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0
  37		(1_var_snapshot)		− __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0

• initial node: 0
• cover edges:

  14	→ 15
  23	→ 3
  28	→ 12
  37	→ 15

• transition edges:

  0	2 1
  1	0 2
  2	1 3
  2	3 4
  3	1 23
  3	3 24
  4	4 5
  5	1 12
  12	6 13
  13	4 14
  13	4 15
  15	1 28
  24	4 37

5.1.1.1.6 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −2.

5.1.1.1.7 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −5.

5.1.1.1.8 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

• version: 1.0