LTS Termination Proof

by T2Cert

Input

• Initial Location: 4
• Transitions: (pre-variables and post-variables)

  0	0	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − i4_0 + n3_0 ≤ 0 ∧ − Fnew5_0 + ans8_post ≤ 0 ∧ Fnew5_0 − ans8_post ≤ 0 ∧ − ans8_post + ret_fib9_post ≤ 0 ∧ ans8_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_post + tmp_post ≤ 0 ∧ ret_fib9_post − tmp_post ≤ 0 ∧ ans8_0 − ans8_post ≤ 0 ∧ − ans8_0 + ans8_post ≤ 0 ∧ ret_fib9_0 − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 ∧ − Fold6_post + Fold6_post ≤ 0 ∧ Fold6_post − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_0 ≤ 0 ∧ Fold6_0 − Fold6_0 ≤ 0 ∧ − Fnew5_post + Fnew5_post ≤ 0 ∧ Fnew5_post − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_0 ≤ 0 ∧ Fnew5_0 − Fnew5_0 ≤ 0
  0	1	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i4_0 − n3_0 ≤ 0 ∧ − Fnew5_0 + temp7_post ≤ 0 ∧ Fnew5_0 − temp7_post ≤ 0 ∧ − Fnew5_0 + Fnew5_post − Fold6_0 ≤ 0 ∧ Fnew5_0 − Fnew5_post + Fold6_0 ≤ 0 ∧ Fold6_post − temp7_post ≤ 0 ∧ − Fold6_post + temp7_post ≤ 0 ∧ −1 − i4_0 + i4_post ≤ 0 ∧ 1 + i4_0 − i4_post ≤ 0 ∧ Fnew5_0 − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_post ≤ 0 ∧ Fold6_0 − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ temp7_0 − temp7_post ≤ 0 ∧ − temp7_0 + temp7_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
  2	2	0:		− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 ∧ − Fold6_post + Fold6_post ≤ 0 ∧ Fold6_post − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_0 ≤ 0 ∧ Fold6_0 − Fold6_0 ≤ 0 ∧ − Fnew5_post + Fnew5_post ≤ 0 ∧ Fnew5_post − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_0 ≤ 0 ∧ Fnew5_0 − Fnew5_0 ≤ 0
  3	3	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ − a_post + n3_post ≤ 0 ∧ a_post − n3_post ≤ 0 ∧ −1 + Fnew5_post ≤ 0 ∧ 1 − Fnew5_post ≤ 0 ∧ Fold6_post ≤ 0 ∧ − Fold6_post ≤ 0 ∧ −2 + i4_post ≤ 0 ∧ 2 − i4_post ≤ 0 ∧ Fnew5_0 − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_post ≤ 0 ∧ Fold6_0 − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_post ≤ 0 ∧ a_0 − a_post ≤ 0 ∧ − a_0 + a_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ n3_0 − n3_post ≤ 0 ∧ − n3_0 + n3_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0
  4	4	3:		− tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 ∧ − Fold6_post + Fold6_post ≤ 0 ∧ Fold6_post − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_0 ≤ 0 ∧ Fold6_0 − Fold6_0 ≤ 0 ∧ − Fnew5_post + Fnew5_post ≤ 0 ∧ Fnew5_post − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_0 ≤ 0 ∧ Fnew5_0 − Fnew5_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)		−30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ −30 + n3_post ≤ 0 ∧ −30 + a_0 ≤ 0 ∧ 30 − a_0 ≤ 0 ∧ −30 + n3_0 ≤ 0
  1		(1)		−30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ −30 + n3_post ≤ 0 ∧ −30 + a_0 ≤ 0 ∧ 30 − a_0 ≤ 0
  2		(2)		−30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ −30 + n3_post ≤ 0 ∧ −30 + a_0 ≤ 0 ∧ 30 − a_0 ≤ 0 ∧ −30 + n3_0 ≤ 0
  3		(3)		TRUE
  4		(4)		TRUE

• initial node: 4
• cover edges:
• transition edges:

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

2 Switch to Cooperation Termination Proof

3 Transition Removal

We remove transitions 0, 3, 4 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* 8 2: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 ∧ − Fold6_post + Fold6_post ≤ 0 ∧ Fold6_post − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_0 ≤ 0 ∧ Fold6_0 − Fold6_0 ≤ 0 ∧ − Fnew5_post + Fnew5_post ≤ 0 ∧ Fnew5_post − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_0 ≤ 0 ∧ Fnew5_0 − Fnew5_0 ≤ 0

5 Location Addition

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

2 6 2_var_snapshot: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 ∧ − Fold6_post + Fold6_post ≤ 0 ∧ Fold6_post − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_0 ≤ 0 ∧ Fold6_0 − Fold6_0 ≤ 0 ∧ − Fnew5_post + Fnew5_post ≤ 0 ∧ Fnew5_post − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_0 ≤ 0 ∧ Fnew5_0 − Fnew5_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 −1021.

6.1.2 Transition Removal

We remove transitions 6, 2 using the following ranking functions, which are bounded by −31.

6.1.3 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by 29.

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