LTS Termination Proof

by T2Cert

Input

Integer Transition System

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 ∧ − ___const_30_0 + ___const_30_0 ≤ 0 ∧ ___const_30_0 − ___const_30_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 ∧ − ___const_30_0 + ___const_30_0 ≤ 0 ∧ ___const_30_0 − ___const_30_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 ∧ − ___const_30_0 + ___const_30_0 ≤ 0 ∧ ___const_30_0 − ___const_30_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 ∧ − ___const_30_0 + a_post ≤ 0 ∧ ___const_30_0 − 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 ∧ − ___const_30_0 + ___const_30_0 ≤ 0 ∧ ___const_30_0 − ___const_30_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 ∧ − ___const_30_0 + ___const_30_0 ≤ 0 ∧ ___const_30_0 − ___const_30_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 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− 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 ∧ − ___const_30_0 + ___const_30_0 ≤ 0 ∧ ___const_30_0 − ___const_30_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

and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 3, 4 using the following ranking functions, which are bounded by −13.

4:	0
3:	0
0:	0
2:	0
1:	0
4:	−5
3:	−6
0:	−7
2:	−7
2_var_snapshot:	−7
2*:	−7
1:	−8

3 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 ∧ − ___const_30_0 + ___const_30_0 ≤ 0 ∧ ___const_30_0 − ___const_30_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

4 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 ∧ − ___const_30_0 + ___const_30_0 ≤ 0 ∧ ___const_30_0 − ___const_30_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 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 { 0, 2, 2_var_snapshot, 2* }.

5.1.1 Transition Removal

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

0:	−1 − 4⋅i4_0 + 4⋅n3_0
2:	1 − 4⋅i4_0 + 4⋅n3_0
2_var_snapshot:	−4⋅i4_0 + 4⋅n3_0
2*:	2 − 4⋅i4_0 + 4⋅n3_0

5.1.2 Transition Removal

We remove transitions 6, 8, 2 using the following ranking functions, which are bounded by −3.

0:	−3
2:	−1
2_var_snapshot:	−2
2*:	0

5.1.3 Splitting Cut-Point Transitions

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

5.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

5.1.3.1.1 Splitting Cut-Point Transitions

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

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