/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 204 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 519 ms]
        (16) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rev1(0, nil) -> 0
   rev1(s(X), nil) -> s(X)
   rev1(X, cons(Y, L)) -> rev1(Y, L)
   rev(nil) -> nil
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
   rev2(X, nil) -> nil
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))

S is empty.
Rewrite Strategy: INNERMOST
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rev1(0', nil) -> 0'
   rev1(s(X), nil) -> s(X)
   rev1(X, cons(Y, L)) -> rev1(Y, L)
   rev(nil) -> nil
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
   rev2(X, nil) -> nil
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
s/0

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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rev1(0', nil) -> 0'
   rev1(s, nil) -> s
   rev1(X, cons(Y, L)) -> rev1(Y, L)
   rev(nil) -> nil
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
   rev2(X, nil) -> nil
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))

S is empty.
Rewrite Strategy: INNERMOST
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
Innermost TRS:
Rules:
rev1(0', nil) -> 0'
rev1(s, nil) -> s
rev1(X, cons(Y, L)) -> rev1(Y, L)
rev(nil) -> nil
rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
rev2(X, nil) -> nil
rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s -> nil:cons -> 0':s
0' :: 0':s
nil :: nil:cons
s :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons

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(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
rev1, rev, rev2

They will be analysed ascendingly in the following order:
rev1 < rev
rev = rev2

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(8)
Obligation:
Innermost TRS:
Rules:
rev1(0', nil) -> 0'
rev1(s, nil) -> s
rev1(X, cons(Y, L)) -> rev1(Y, L)
rev(nil) -> nil
rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
rev2(X, nil) -> nil
rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s -> nil:cons -> 0':s
0' :: 0':s
nil :: nil:cons
s :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))


The following defined symbols remain to be analysed:
rev1, rev, rev2

They will be analysed ascendingly in the following order:
rev1 < rev
rev = rev2

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rev1(0', gen_nil:cons3_0(n5_0)) -> 0', rt in Omega(1 + n5_0)

Induction Base:
rev1(0', gen_nil:cons3_0(0)) ->_R^Omega(1)
0'

Induction Step:
rev1(0', gen_nil:cons3_0(+(n5_0, 1))) ->_R^Omega(1)
rev1(0', gen_nil:cons3_0(n5_0)) ->_IH
0'

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(10)
Complex Obligation (BEST)

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(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
rev1(0', nil) -> 0'
rev1(s, nil) -> s
rev1(X, cons(Y, L)) -> rev1(Y, L)
rev(nil) -> nil
rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
rev2(X, nil) -> nil
rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s -> nil:cons -> 0':s
0' :: 0':s
nil :: nil:cons
s :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))


The following defined symbols remain to be analysed:
rev1, rev, rev2

They will be analysed ascendingly in the following order:
rev1 < rev
rev = rev2

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(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(14)
Obligation:
Innermost TRS:
Rules:
rev1(0', nil) -> 0'
rev1(s, nil) -> s
rev1(X, cons(Y, L)) -> rev1(Y, L)
rev(nil) -> nil
rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
rev2(X, nil) -> nil
rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s -> nil:cons -> 0':s
0' :: 0':s
nil :: nil:cons
s :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons


Lemmas:
rev1(0', gen_nil:cons3_0(n5_0)) -> 0', rt in Omega(1 + n5_0)


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))


The following defined symbols remain to be analysed:
rev2, rev

They will be analysed ascendingly in the following order:
rev = rev2

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rev2(0', gen_nil:cons3_0(+(1, n203_0))) -> *4_0, rt in Omega(n203_0)

Induction Base:
rev2(0', gen_nil:cons3_0(+(1, 0)))

Induction Step:
rev2(0', gen_nil:cons3_0(+(1, +(n203_0, 1)))) ->_R^Omega(1)
rev(cons(0', rev(rev2(0', gen_nil:cons3_0(+(1, n203_0)))))) ->_IH
rev(cons(0', rev(*4_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(16)
Obligation:
Innermost TRS:
Rules:
rev1(0', nil) -> 0'
rev1(s, nil) -> s
rev1(X, cons(Y, L)) -> rev1(Y, L)
rev(nil) -> nil
rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
rev2(X, nil) -> nil
rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s -> nil:cons -> 0':s
0' :: 0':s
nil :: nil:cons
s :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons


Lemmas:
rev1(0', gen_nil:cons3_0(n5_0)) -> 0', rt in Omega(1 + n5_0)
rev2(0', gen_nil:cons3_0(+(1, n203_0))) -> *4_0, rt in Omega(n203_0)


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))


The following defined symbols remain to be analysed:
rev

They will be analysed ascendingly in the following order:
rev = rev2