LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

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: −30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ −30 + n3_post ≤ 0 ∧ −30 + a_0 ≤ 0 ∧ 30 − a_0 ≤ 0
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: TRUE
4: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 5 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
and for every transition t, a duplicate t is considered.

3 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

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.

0: − a_0 − 33⋅i4_0
2: 1 − 33⋅i4_0
2_var_snapshot: −33⋅i4_0
2*: 2 − 33⋅i4_0

6.1.2 Transition Removal

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

0: − a_0 − a_post
2: 0
2_var_snapshot: − a_post
2*: a_post

6.1.3 Transition Removal

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

0: 0
2: 0
2_var_snapshot: 0
2*: a_0

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

Here we consider cut-point transition 5.

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

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