WORST_CASE(Omega(n^1), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^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), 356 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) 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:

   g(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldf(x, nil) -> x
   foldf(x, cons(y, z)) -> f(foldf(x, z), y)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, cons(C, c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b))
   f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c)

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:

   g(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldf(x, nil) -> x
   foldf(x, cons(y, z)) -> f(foldf(x, z), y)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, cons(C, c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b))
   f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c)

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

   g(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldf(x, nil) -> x
   foldf(x, cons(y, z)) -> f(foldf(x, z), y)
   f(t, x) -> f'(t, g(x))
   f'(triple(b, c), C) -> triple(b, cons(C, c))
   f'(triple(b, c), B) -> f(triple(b, c), A)
   f'(triple(b, c), A) -> f''(foldf(triple(nil, c), b))
   f''(triple(b, c)) -> foldf(triple(b, nil), c)

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:
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldf(x, nil) -> x
foldf(x, cons(y, z)) -> f(foldf(x, z), y)
f(t, x) -> f'(t, g(x))
f'(triple(b, c), C) -> triple(b, cons(C, c))
f'(triple(b, c), B) -> f(triple(b, c), A)
f'(triple(b, c), A) -> f''(foldf(triple(nil, c), b))
f''(triple(b, c)) -> foldf(triple(b, nil), c)

Types:
g :: A:B:C -> A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple -> nil:cons -> triple
nil :: nil:cons
cons :: A:B:C -> nil:cons -> nil:cons
f :: triple -> A:B:C -> triple
f' :: triple -> A:B:C -> triple
triple :: nil:cons -> nil:cons -> triple
f'' :: triple -> triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat -> nil:cons

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(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
foldf, f, f', f''

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

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(8)
Obligation:
Innermost TRS:
Rules:
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldf(x, nil) -> x
foldf(x, cons(y, z)) -> f(foldf(x, z), y)
f(t, x) -> f'(t, g(x))
f'(triple(b, c), C) -> triple(b, cons(C, c))
f'(triple(b, c), B) -> f(triple(b, c), A)
f'(triple(b, c), A) -> f''(foldf(triple(nil, c), b))
f''(triple(b, c)) -> foldf(triple(b, nil), c)

Types:
g :: A:B:C -> A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple -> nil:cons -> triple
nil :: nil:cons
cons :: A:B:C -> nil:cons -> nil:cons
f :: triple -> A:B:C -> triple
f' :: triple -> A:B:C -> triple
triple :: nil:cons -> nil:cons -> triple
f'' :: triple -> triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat -> nil:cons


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


The following defined symbols remain to be analysed:
f, foldf, f', f''

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
foldf(triple(nil, nil), gen_nil:cons4_0(n153_0)) -> triple(gen_nil:cons4_0(0), gen_nil:cons4_0(0)), rt in Omega(1 + n153_0)

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

Induction Step:
foldf(triple(nil, nil), gen_nil:cons4_0(+(n153_0, 1))) ->_R^Omega(1)
f(foldf(triple(nil, nil), gen_nil:cons4_0(n153_0)), A) ->_IH
f(triple(gen_nil:cons4_0(0), gen_nil:cons4_0(0)), A) ->_R^Omega(1)
f'(triple(gen_nil:cons4_0(0), gen_nil:cons4_0(0)), g(A)) ->_R^Omega(1)
f'(triple(gen_nil:cons4_0(0), gen_nil:cons4_0(0)), A) ->_R^Omega(1)
f''(foldf(triple(nil, gen_nil:cons4_0(0)), gen_nil:cons4_0(0))) ->_R^Omega(1)
f''(triple(nil, gen_nil:cons4_0(0))) ->_R^Omega(1)
foldf(triple(nil, nil), gen_nil:cons4_0(0)) ->_R^Omega(1)
triple(nil, nil)

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:
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldf(x, nil) -> x
foldf(x, cons(y, z)) -> f(foldf(x, z), y)
f(t, x) -> f'(t, g(x))
f'(triple(b, c), C) -> triple(b, cons(C, c))
f'(triple(b, c), B) -> f(triple(b, c), A)
f'(triple(b, c), A) -> f''(foldf(triple(nil, c), b))
f''(triple(b, c)) -> foldf(triple(b, nil), c)

Types:
g :: A:B:C -> A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple -> nil:cons -> triple
nil :: nil:cons
cons :: A:B:C -> nil:cons -> nil:cons
f :: triple -> A:B:C -> triple
f' :: triple -> A:B:C -> triple
triple :: nil:cons -> nil:cons -> triple
f'' :: triple -> triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat -> nil:cons


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


The following defined symbols remain to be analysed:
foldf

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

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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:
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldf(x, nil) -> x
foldf(x, cons(y, z)) -> f(foldf(x, z), y)
f(t, x) -> f'(t, g(x))
f'(triple(b, c), C) -> triple(b, cons(C, c))
f'(triple(b, c), B) -> f(triple(b, c), A)
f'(triple(b, c), A) -> f''(foldf(triple(nil, c), b))
f''(triple(b, c)) -> foldf(triple(b, nil), c)

Types:
g :: A:B:C -> A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple -> nil:cons -> triple
nil :: nil:cons
cons :: A:B:C -> nil:cons -> nil:cons
f :: triple -> A:B:C -> triple
f' :: triple -> A:B:C -> triple
triple :: nil:cons -> nil:cons -> triple
f'' :: triple -> triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat -> nil:cons


Lemmas:
foldf(triple(nil, nil), gen_nil:cons4_0(n153_0)) -> triple(gen_nil:cons4_0(0), gen_nil:cons4_0(0)), rt in Omega(1 + n153_0)


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


The following defined symbols remain to be analysed:
f, f', f''

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''