/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(Omega(n^3), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^3, 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), 308 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), 89 ms]
        (16) BEST
            (17) proven lower bound
                (18) LowerBoundPropagationProof [FINISHED, 0 ms]
                (19) BOUNDS(n^3, INF)
            (20) typed CpxTrs
                (21) RewriteLemmaProof [LOWER BOUND(ID), 544 ms]
                (22) BOUNDS(1, INF)


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   power(x', Cons(x, xs)) -> mult(x', power(x', xs))
   mult(x', Cons(x, xs)) -> add0(x', mult(x', xs))
   add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs))
   power(x, Nil) -> Cons(Nil, Nil)
   mult(x, Nil) -> Nil
   add0(x, Nil) -> x
   goal(x, y) -> power(x, y)

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^3, INF).


The TRS R consists of the following rules:

   power(x', Cons(x, xs)) -> mult(x', power(x', xs))
   mult(x', Cons(x, xs)) -> add0(x', mult(x', xs))
   add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs))
   power(x, Nil) -> Cons(Nil, Nil)
   mult(x, Nil) -> Nil
   add0(x, Nil) -> x
   goal(x, y) -> power(x, y)

S is empty.
Rewrite Strategy: INNERMOST
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(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0

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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   power(x', Cons(xs)) -> mult(x', power(x', xs))
   mult(x', Cons(xs)) -> add0(x', mult(x', xs))
   add0(x', Cons(xs)) -> Cons(add0(x', xs))
   power(x, Nil) -> Cons(Nil)
   mult(x, Nil) -> Nil
   add0(x, Nil) -> x
   goal(x, y) -> power(x, y)

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:
power(x', Cons(xs)) -> mult(x', power(x', xs))
mult(x', Cons(xs)) -> add0(x', mult(x', xs))
add0(x', Cons(xs)) -> Cons(add0(x', xs))
power(x, Nil) -> Cons(Nil)
mult(x, Nil) -> Nil
add0(x, Nil) -> x
goal(x, y) -> power(x, y)

Types:
power :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
mult :: Cons:Nil -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil

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(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
power, mult, add0

They will be analysed ascendingly in the following order:
mult < power
add0 < mult

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(8)
Obligation:
Innermost TRS:
Rules:
power(x', Cons(xs)) -> mult(x', power(x', xs))
mult(x', Cons(xs)) -> add0(x', mult(x', xs))
add0(x', Cons(xs)) -> Cons(add0(x', xs))
power(x, Nil) -> Cons(Nil)
mult(x, Nil) -> Nil
add0(x, Nil) -> x
goal(x, y) -> power(x, y)

Types:
power :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
mult :: Cons:Nil -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
add0, power, mult

They will be analysed ascendingly in the following order:
mult < power
add0 < mult

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) -> gen_Cons:Nil2_1(+(n4_1, a)), rt in Omega(1 + n4_1)

Induction Base:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(0)) ->_R^Omega(1)
gen_Cons:Nil2_1(a)

Induction Step:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(n4_1, 1))) ->_R^Omega(1)
Cons(add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1))) ->_IH
Cons(gen_Cons:Nil2_1(+(a, c5_1)))

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:
power(x', Cons(xs)) -> mult(x', power(x', xs))
mult(x', Cons(xs)) -> add0(x', mult(x', xs))
add0(x', Cons(xs)) -> Cons(add0(x', xs))
power(x, Nil) -> Cons(Nil)
mult(x, Nil) -> Nil
add0(x, Nil) -> x
goal(x, y) -> power(x, y)

Types:
power :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
mult :: Cons:Nil -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
add0, power, mult

They will be analysed ascendingly in the following order:
mult < power
add0 < mult

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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:
power(x', Cons(xs)) -> mult(x', power(x', xs))
mult(x', Cons(xs)) -> add0(x', mult(x', xs))
add0(x', Cons(xs)) -> Cons(add0(x', xs))
power(x, Nil) -> Cons(Nil)
mult(x, Nil) -> Nil
add0(x, Nil) -> x
goal(x, y) -> power(x, y)

Types:
power :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
mult :: Cons:Nil -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Lemmas:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) -> gen_Cons:Nil2_1(+(n4_1, a)), rt in Omega(1 + n4_1)


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
mult, power

They will be analysed ascendingly in the following order:
mult < power

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n504_1)) -> gen_Cons:Nil2_1(*(n504_1, a)), rt in Omega(1 + a*n504_1^2 + n504_1)

Induction Base:
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(0)) ->_R^Omega(1)
Nil

Induction Step:
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(n504_1, 1))) ->_R^Omega(1)
add0(gen_Cons:Nil2_1(a), mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n504_1))) ->_IH
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(*(c505_1, a))) ->_L^Omega(1 + a*n504_1)
gen_Cons:Nil2_1(+(*(n504_1, a), a))

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
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(16)
Complex Obligation (BEST)

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(17)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
power(x', Cons(xs)) -> mult(x', power(x', xs))
mult(x', Cons(xs)) -> add0(x', mult(x', xs))
add0(x', Cons(xs)) -> Cons(add0(x', xs))
power(x, Nil) -> Cons(Nil)
mult(x, Nil) -> Nil
add0(x, Nil) -> x
goal(x, y) -> power(x, y)

Types:
power :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
mult :: Cons:Nil -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Lemmas:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) -> gen_Cons:Nil2_1(+(n4_1, a)), rt in Omega(1 + n4_1)


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
mult, power

They will be analysed ascendingly in the following order:
mult < power

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(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(19)
BOUNDS(n^3, INF)

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(20)
Obligation:
Innermost TRS:
Rules:
power(x', Cons(xs)) -> mult(x', power(x', xs))
mult(x', Cons(xs)) -> add0(x', mult(x', xs))
add0(x', Cons(xs)) -> Cons(add0(x', xs))
power(x, Nil) -> Cons(Nil)
mult(x, Nil) -> Nil
add0(x, Nil) -> x
goal(x, y) -> power(x, y)

Types:
power :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
mult :: Cons:Nil -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Lemmas:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) -> gen_Cons:Nil2_1(+(n4_1, a)), rt in Omega(1 + n4_1)
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n504_1)) -> gen_Cons:Nil2_1(*(n504_1, a)), rt in Omega(1 + a*n504_1^2 + n504_1)


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
power
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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
power(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(1, n1128_1))) -> *3_1, rt in Omega(n1128_1)

Induction Base:
power(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(1, 0)))

Induction Step:
power(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(1, +(n1128_1, 1)))) ->_R^Omega(1)
mult(gen_Cons:Nil2_1(a), power(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(1, n1128_1)))) ->_IH
mult(gen_Cons:Nil2_1(a), *3_1)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(22)
BOUNDS(1, INF)