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 ∧ − x9 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ −1 − 2⋅x10 + x9 ≤ 0 ∧ 1 + 2⋅x10 − x9 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − x22 + x22 ≤ 0 ∧ x22 − x22 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x14 + x14 ≤ 0 ∧ x14 − x14 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 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
  1	1	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − x13 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ −1 + x13 − 2⋅x14 ≤ 0 ∧ 1 − x13 + 2⋅x14 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ −1 + x13 − 2⋅x14 ≤ 0 ∧ − x13 + 2⋅x14 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ −1 + arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x9 + x9 ≤ 0 ∧ x9 − x9 ≤ 0 ∧ − x22 + x22 ≤ 0 ∧ x22 − x22 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x10 + x10 ≤ 0 ∧ x10 − x10 ≤ 0
  0	2	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − x17 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ x17 − 2⋅x18 ≤ 0 ∧ − x17 + 2⋅x18 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − x9 + x9 ≤ 0 ∧ x9 − x9 ≤ 0 ∧ − x22 + x22 ≤ 0 ∧ x22 − x22 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x14 + x14 ≤ 0 ∧ x14 − x14 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x10 + x10 ≤ 0 ∧ x10 − x10 ≤ 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
  1	3	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − x21 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ x21 − 2⋅x22 ≤ 0 ∧ − x21 + 2⋅x22 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ −1 + x21 − 2⋅x22 ≤ 0 ∧ − x21 + 2⋅x22 ≤ 0 ∧ − arg1P ≤ 0 ∧ arg1P ≤ 0 ∧ 1 − arg2P ≤ 0 ∧ −1 + arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x9 + x9 ≤ 0 ∧ x9 − x9 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x14 + x14 ≤ 0 ∧ x14 − x14 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x10 + x10 ≤ 0 ∧ x10 − x10 ≤ 0
  2	4	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ arg1 − arg2 ≤ 0 ∧ − arg1 + arg2 ≤ 0 ∧ −1 − arg1P + arg1 ≤ 0 ∧ 1 + arg1P − arg1 ≤ 0 ∧ −1 + arg1 − arg2P ≤ 0 ∧ 1 − arg1 + arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x9 + x9 ≤ 0 ∧ x9 − x9 ≤ 0 ∧ − x22 + x22 ≤ 0 ∧ x22 − x22 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x14 + x14 ≤ 0 ∧ x14 − x14 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x10 + x10 ≤ 0 ∧ x10 − x10 ≤ 0
  3	5	0:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − x9 + x9 ≤ 0 ∧ x9 − x9 ≤ 0 ∧ − x22 + x22 ≤ 0 ∧ x22 − x22 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x14 + x14 ≤ 0 ∧ x14 − x14 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x10 + x10 ≤ 0 ∧ x10 − x10 ≤ 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)		1 − arg1 ≤ 0 ∧ 1 − arg2 ≤ 0
  2		(2)		TRUE
  3		(3)		TRUE

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

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

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 1, 2, 3, 5 using the following ranking functions, which are bounded by −13.

4 Location Addition

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

2* 9 2: − x9 + x9 ≤ 0 ∧ x9 − x9 ≤ 0 ∧ − x22 + x22 ≤ 0 ∧ x22 − x22 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x14 + x14 ≤ 0 ∧ x14 − x14 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x10 + x10 ≤ 0 ∧ x10 − x10 ≤ 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.

2 7 2_var_snapshot: − x9 + x9 ≤ 0 ∧ x9 − x9 ≤ 0 ∧ − x22 + x22 ≤ 0 ∧ x22 − x22 ≤ 0 ∧ − x21 + x21 ≤ 0 ∧ x21 − x21 ≤ 0 ∧ − x18 + x18 ≤ 0 ∧ x18 − x18 ≤ 0 ∧ − x17 + x17 ≤ 0 ∧ x17 − x17 ≤ 0 ∧ − x14 + x14 ≤ 0 ∧ x14 − x14 ≤ 0 ∧ − x13 + x13 ≤ 0 ∧ x13 − x13 ≤ 0 ∧ − x10 + x10 ≤ 0 ∧ x10 − x10 ≤ 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 { 2, 2_var_snapshot, 2* }.

6.1.1 Splitting Cut-Point Transitions

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

6.1.1.1 Cut-Point Subproblem 1/1

6.1.1.1.1 Invariant Updates

The following invariants are asserted.

The invariants are proved as follows.

IMPACT Invariant Proof

• nodes (location) invariant:

  0		(3)		TRUE
  1		(0)		TRUE
  2		(1)		1 − arg1 ≤ 0 ∧ 1 − arg2 ≤ 0
  3		(1)		1 − arg1 ≤ 0 ∧ 1 − arg2 ≤ 0
  4		(2)		1 − arg1 + arg2 ≤ 0
  5		(2)		1 + arg1 − arg2 ≤ 0
  6		(2)		1 ≤ 0
  7		(2)		1 + arg1 − arg2 ≤ 0
  8		(2_var_snapshot)		1 + arg1 − arg2 ≤ 0
  15		(2*)		1 ≤ 0
  22		(2)		1 ≤ 0
  23		(2)		1 − arg1 + arg2 ≤ 0
  24		(2_var_snapshot)		1 − arg1 + arg2 ≤ 0
  29		(2*)		1 ≤ 0

• initial node: 0
• cover edges:

  2

  → 3

• transition edges:

  0	5 1
  1	0 2
  1	2 3
  3	1 4
  3	3 5
  4	4 22
  4	6 23
  5	4 6
  5	6 7
  7	7 8
  8	4 15
  23	7 24
  24	4 29

6.1.1.1.2 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −5.

6.1.1.1.3 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

• version: 1.0