/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(EXP, 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), 205 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [FINISHED, 81 ms]
        (16) BOUNDS(EXP, INF)


----------------------------------------

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


The TRS R consists of the following rules:

   empty(nil) -> true
   empty(cons(x, l)) -> false
   head(cons(x, l)) -> x
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   rev(nil) -> nil
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
   last(x, l) -> if(empty(l), x, l)
   if(true, x, l) -> x
   if(false, x, l) -> last(head(l), tail(l))
   rev2(x, nil) -> nil
   rev2(x, cons(y, l)) -> rev(cons(x, 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(EXP, INF).


The TRS R consists of the following rules:

   empty(nil) -> true
   empty(cons(x, l)) -> false
   head(cons(x, l)) -> x
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   rev(nil) -> nil
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
   last(x, l) -> if(empty(l), x, l)
   if(true, x, l) -> x
   if(false, x, l) -> last(head(l), tail(l))
   rev2(x, nil) -> nil
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
rev1/0
rev1/1

----------------------------------------

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


The TRS R consists of the following rules:

   empty(nil) -> true
   empty(cons(x, l)) -> false
   head(cons(x, l)) -> x
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   rev(nil) -> nil
   rev(cons(x, l)) -> cons(rev1, rev2(x, l))
   last(x, l) -> if(empty(l), x, l)
   if(true, x, l) -> x
   if(false, x, l) -> last(head(l), tail(l))
   rev2(x, nil) -> nil
   rev2(x, cons(y, l)) -> rev(cons(x, 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:
empty(nil) -> true
empty(cons(x, l)) -> false
head(cons(x, l)) -> x
tail(nil) -> nil
tail(cons(x, l)) -> l
rev(nil) -> nil
rev(cons(x, l)) -> cons(rev1, rev2(x, l))
last(x, l) -> if(empty(l), x, l)
if(true, x, l) -> x
if(false, x, l) -> last(head(l), tail(l))
rev2(x, nil) -> nil
rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: rev1 -> nil:cons -> nil:cons
false :: true:false
head :: nil:cons -> rev1
tail :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: rev1
rev2 :: rev1 -> nil:cons -> nil:cons
last :: rev1 -> nil:cons -> rev1
if :: true:false -> rev1 -> nil:cons -> rev1
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_rev13_0 :: rev1
gen_nil:cons4_0 :: Nat -> nil:cons

----------------------------------------

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

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

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(8)
Obligation:
Innermost TRS:
Rules:
empty(nil) -> true
empty(cons(x, l)) -> false
head(cons(x, l)) -> x
tail(nil) -> nil
tail(cons(x, l)) -> l
rev(nil) -> nil
rev(cons(x, l)) -> cons(rev1, rev2(x, l))
last(x, l) -> if(empty(l), x, l)
if(true, x, l) -> x
if(false, x, l) -> last(head(l), tail(l))
rev2(x, nil) -> nil
rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: rev1 -> nil:cons -> nil:cons
false :: true:false
head :: nil:cons -> rev1
tail :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: rev1
rev2 :: rev1 -> nil:cons -> nil:cons
last :: rev1 -> nil:cons -> rev1
if :: true:false -> rev1 -> nil:cons -> rev1
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_rev13_0 :: rev1
gen_nil:cons4_0 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(rev1, gen_nil:cons4_0(x))


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

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

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
last(rev1, gen_nil:cons4_0(n6_0)) -> rev1, rt in Omega(1 + n6_0)

Induction Base:
last(rev1, gen_nil:cons4_0(0)) ->_R^Omega(1)
if(empty(gen_nil:cons4_0(0)), rev1, gen_nil:cons4_0(0)) ->_R^Omega(1)
if(true, rev1, gen_nil:cons4_0(0)) ->_R^Omega(1)
rev1

Induction Step:
last(rev1, gen_nil:cons4_0(+(n6_0, 1))) ->_R^Omega(1)
if(empty(gen_nil:cons4_0(+(n6_0, 1))), rev1, gen_nil:cons4_0(+(n6_0, 1))) ->_R^Omega(1)
if(false, rev1, gen_nil:cons4_0(+(1, n6_0))) ->_R^Omega(1)
last(head(gen_nil:cons4_0(+(1, n6_0))), tail(gen_nil:cons4_0(+(1, n6_0)))) ->_R^Omega(1)
last(rev1, tail(gen_nil:cons4_0(+(1, n6_0)))) ->_R^Omega(1)
last(rev1, gen_nil:cons4_0(n6_0)) ->_IH
rev1

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:
empty(nil) -> true
empty(cons(x, l)) -> false
head(cons(x, l)) -> x
tail(nil) -> nil
tail(cons(x, l)) -> l
rev(nil) -> nil
rev(cons(x, l)) -> cons(rev1, rev2(x, l))
last(x, l) -> if(empty(l), x, l)
if(true, x, l) -> x
if(false, x, l) -> last(head(l), tail(l))
rev2(x, nil) -> nil
rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: rev1 -> nil:cons -> nil:cons
false :: true:false
head :: nil:cons -> rev1
tail :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: rev1
rev2 :: rev1 -> nil:cons -> nil:cons
last :: rev1 -> nil:cons -> rev1
if :: true:false -> rev1 -> nil:cons -> rev1
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_rev13_0 :: rev1
gen_nil:cons4_0 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(rev1, gen_nil:cons4_0(x))


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

They will be analysed ascendingly in the following order:
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:
empty(nil) -> true
empty(cons(x, l)) -> false
head(cons(x, l)) -> x
tail(nil) -> nil
tail(cons(x, l)) -> l
rev(nil) -> nil
rev(cons(x, l)) -> cons(rev1, rev2(x, l))
last(x, l) -> if(empty(l), x, l)
if(true, x, l) -> x
if(false, x, l) -> last(head(l), tail(l))
rev2(x, nil) -> nil
rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: rev1 -> nil:cons -> nil:cons
false :: true:false
head :: nil:cons -> rev1
tail :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: rev1
rev2 :: rev1 -> nil:cons -> nil:cons
last :: rev1 -> nil:cons -> rev1
if :: true:false -> rev1 -> nil:cons -> rev1
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_rev13_0 :: rev1
gen_nil:cons4_0 :: Nat -> nil:cons


Lemmas:
last(rev1, gen_nil:cons4_0(n6_0)) -> rev1, rt in Omega(1 + n6_0)


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(rev1, gen_nil:cons4_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 (FINISHED)
Proved the following rewrite lemma:
rev2(rev1, gen_nil:cons4_0(n341_0)) -> gen_nil:cons4_0(n341_0), rt in Omega(EXP)

Induction Base:
rev2(rev1, gen_nil:cons4_0(0)) ->_R^Omega(1)
nil

Induction Step:
rev2(rev1, gen_nil:cons4_0(+(n341_0, 1))) ->_R^Omega(1)
rev(cons(rev1, rev2(rev1, gen_nil:cons4_0(n341_0)))) ->_IH
rev(cons(rev1, gen_nil:cons4_0(c342_0))) ->_R^Omega(1)
cons(rev1, rev2(rev1, gen_nil:cons4_0(n341_0))) ->_IH
cons(rev1, gen_nil:cons4_0(c342_0))

We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP
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(16)
BOUNDS(EXP, INF)