WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 180 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 320 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 2 ms]
        (18) BOUNDS(1, INF)


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


The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> z

S is empty.
Rewrite Strategy: INNERMOST
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(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))


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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> z

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> z

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
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(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> z

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Innermost TRS:
Rules:
f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z
encArg(a) -> a
encArg(g(x_1)) -> g(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_a -> a
encode_g(x_1) -> g(encArg(x_1))
encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
a :: a:g:cons_f:cons_h
g :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encArg :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
cons_f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
cons_h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_a :: a:g:cons_f:cons_h
encode_g :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
hole_a:g:cons_f:cons_h1_0 :: a:g:cons_f:cons_h
gen_a:g:cons_f:cons_h2_0 :: Nat -> a:g:cons_f:cons_h

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, h, encArg

They will be analysed ascendingly in the following order:
f = h
f < encArg
h < encArg

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(10)
Obligation:
Innermost TRS:
Rules:
f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z
encArg(a) -> a
encArg(g(x_1)) -> g(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_a -> a
encode_g(x_1) -> g(encArg(x_1))
encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
a :: a:g:cons_f:cons_h
g :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encArg :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
cons_f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
cons_h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_a :: a:g:cons_f:cons_h
encode_g :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
hole_a:g:cons_f:cons_h1_0 :: a:g:cons_f:cons_h
gen_a:g:cons_f:cons_h2_0 :: Nat -> a:g:cons_f:cons_h


Generator Equations:
gen_a:g:cons_f:cons_h2_0(0) <=> a
gen_a:g:cons_f:cons_h2_0(+(x, 1)) <=> g(gen_a:g:cons_f:cons_h2_0(x))


The following defined symbols remain to be analysed:
h, f, encArg

They will be analysed ascendingly in the following order:
f = h
f < encArg
h < encArg

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
h(gen_a:g:cons_f:cons_h2_0(n4_0), gen_a:g:cons_f:cons_h2_0(0), gen_a:g:cons_f:cons_h2_0(1)) -> gen_a:g:cons_f:cons_h2_0(+(1, n4_0)), rt in Omega(1 + n4_0)

Induction Base:
h(gen_a:g:cons_f:cons_h2_0(0), gen_a:g:cons_f:cons_h2_0(0), gen_a:g:cons_f:cons_h2_0(1)) ->_R^Omega(1)
gen_a:g:cons_f:cons_h2_0(1)

Induction Step:
h(gen_a:g:cons_f:cons_h2_0(+(n4_0, 1)), gen_a:g:cons_f:cons_h2_0(0), gen_a:g:cons_f:cons_h2_0(1)) ->_R^Omega(1)
f(gen_a:g:cons_f:cons_h2_0(0), h(gen_a:g:cons_f:cons_h2_0(n4_0), gen_a:g:cons_f:cons_h2_0(0), gen_a:g:cons_f:cons_h2_0(1))) ->_IH
f(gen_a:g:cons_f:cons_h2_0(0), gen_a:g:cons_f:cons_h2_0(+(1, c5_0))) ->_R^Omega(1)
g(g(gen_a:g:cons_f:cons_h2_0(n4_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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(12)
Complex Obligation (BEST)

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(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z
encArg(a) -> a
encArg(g(x_1)) -> g(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_a -> a
encode_g(x_1) -> g(encArg(x_1))
encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
a :: a:g:cons_f:cons_h
g :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encArg :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
cons_f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
cons_h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_a :: a:g:cons_f:cons_h
encode_g :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
hole_a:g:cons_f:cons_h1_0 :: a:g:cons_f:cons_h
gen_a:g:cons_f:cons_h2_0 :: Nat -> a:g:cons_f:cons_h


Generator Equations:
gen_a:g:cons_f:cons_h2_0(0) <=> a
gen_a:g:cons_f:cons_h2_0(+(x, 1)) <=> g(gen_a:g:cons_f:cons_h2_0(x))


The following defined symbols remain to be analysed:
h, f, encArg

They will be analysed ascendingly in the following order:
f = h
f < encArg
h < encArg

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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Innermost TRS:
Rules:
f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z
encArg(a) -> a
encArg(g(x_1)) -> g(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_a -> a
encode_g(x_1) -> g(encArg(x_1))
encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
a :: a:g:cons_f:cons_h
g :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encArg :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
cons_f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
cons_h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_f :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_a :: a:g:cons_f:cons_h
encode_g :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h
encode_h :: a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h -> a:g:cons_f:cons_h
hole_a:g:cons_f:cons_h1_0 :: a:g:cons_f:cons_h
gen_a:g:cons_f:cons_h2_0 :: Nat -> a:g:cons_f:cons_h


Lemmas:
h(gen_a:g:cons_f:cons_h2_0(n4_0), gen_a:g:cons_f:cons_h2_0(0), gen_a:g:cons_f:cons_h2_0(1)) -> gen_a:g:cons_f:cons_h2_0(+(1, n4_0)), rt in Omega(1 + n4_0)


Generator Equations:
gen_a:g:cons_f:cons_h2_0(0) <=> a
gen_a:g:cons_f:cons_h2_0(+(x, 1)) <=> g(gen_a:g:cons_f:cons_h2_0(x))


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

They will be analysed ascendingly in the following order:
f = h
f < encArg
h < encArg

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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_a:g:cons_f:cons_h2_0(n1816_0)) -> gen_a:g:cons_f:cons_h2_0(n1816_0), rt in Omega(0)

Induction Base:
encArg(gen_a:g:cons_f:cons_h2_0(0)) ->_R^Omega(0)
a

Induction Step:
encArg(gen_a:g:cons_f:cons_h2_0(+(n1816_0, 1))) ->_R^Omega(0)
g(encArg(gen_a:g:cons_f:cons_h2_0(n1816_0))) ->_IH
g(gen_a:g:cons_f:cons_h2_0(c1817_0))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(18)
BOUNDS(1, INF)