LTS Termination Proof

by T2Cert

Input

• Initial Location: 10
• Transitions: (pre-variables and post-variables)

  0	0	1:		1 − i_0 + n_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  0	1	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_0 − n_0 ≤ 0 ∧ −1 − i_0 + i_post ≤ 0 ∧ 1 + i_0 − i_post ≤ 0 ∧ −2 − j_0 + j_post ≤ 0 ∧ 2 + j_0 − j_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ j_0 − j_post ≤ 0 ∧ − j_0 + j_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  3	2	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ i_post ≤ 0 ∧ − i_post ≤ 0 ∧ i_0 − i_post ≤ 0 ∧ − i_0 + i_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  4	3	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ n_post ≤ 0 ∧ − n_post ≤ 0 ∧ n_0 − n_post ≤ 0 ∧ − n_0 + n_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  5	4	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ tmp_0 ≤ 0 ∧ − tmp_0 ≤ 0 ∧ − ___const_1023_0 + n_post ≤ 0 ∧ ___const_1023_0 − n_post ≤ 0 ∧ n_0 − n_post ≤ 0 ∧ − n_0 + n_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  5	5	4:		1 − tmp_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  5	6	4:		1 + tmp_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  2	7	0:		− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  6	8	7:		− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  8	9	6:		___const_1023_0 − b_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  8	10	6:		1 − ___const_1023_0 + b_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  1	11	6:		1 + b_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  1	12	8:		− b_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  9	13	5:		0 ≤ 0 ∧ 0 ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0
  10	14	9:		− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 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)		TRUE
  3		(3)		TRUE
  4		(4)		TRUE
  5		(5)		TRUE
  6		(6)		TRUE
  7		(7)		TRUE
  8		(8)		− b_0 ≤ 0
  9		(9)		TRUE
  10		(10)		TRUE

• initial node: 10
• cover edges:
• transition edges:

  0	0 1
  0	1 2
  1	11 6
  1	12 8
  2	7 0
  3	2 2
  4	3 3
  5	4 3
  5	5 4
  5	6 4
  6	8 7
  8	9 6
  8	10 6
  9	13 5
  10	14 9

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14 using the following ranking functions, which are bounded by −25.

4 Location Addition

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

2* 18 2: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_0 ≤ 0

5 Location Addition

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

2 16 2_var_snapshot: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − n_post + n_post ≤ 0 ∧ n_post − n_post ≤ 0 ∧ − n_0 + n_0 ≤ 0 ∧ n_0 − n_0 ≤ 0 ∧ − j_post + j_post ≤ 0 ∧ j_post − j_post ≤ 0 ∧ − j_0 + j_0 ≤ 0 ∧ j_0 − j_0 ≤ 0 ∧ − i_post + i_post ≤ 0 ∧ i_post − i_post ≤ 0 ∧ − i_0 + i_0 ≤ 0 ∧ i_0 − i_0 ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − ___const_1023_0 + ___const_1023_0 ≤ 0 ∧ ___const_1023_0 − ___const_1023_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, 2, 2_var_snapshot, 2* }.

6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −2.

6.1.2 Transition Removal

We remove transitions 16, 18 using the following ranking functions, which are bounded by −1.

6.1.3 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −1.

6.1.4 Splitting Cut-Point Transitions

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

6.1.4.1 Cut-Point Subproblem 1/1

6.1.4.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert

• version: 1.0