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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 118 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), 910 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), 253 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:

   ack(0, y) -> s(y)
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   f(s(x), y) -> f(x, s(x))
   f(x, s(y)) -> f(y, x)
   f(x, y) -> ack(x, y)
   ack(s(x), y) -> f(x, x)

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(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))


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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:

   ack(0, y) -> s(y)
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   f(s(x), y) -> f(x, s(x))
   f(x, s(y)) -> f(y, x)
   f(x, y) -> ack(x, y)
   ack(s(x), y) -> f(x, x)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

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:

   ack(0, y) -> s(y)
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   f(s(x), y) -> f(x, s(x))
   f(x, s(y)) -> f(y, x)
   f(x, y) -> ack(x, y)
   ack(s(x), y) -> f(x, x)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(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:

   ack(0', y) -> s(y)
   ack(s(x), 0') -> ack(x, s(0'))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   f(s(x), y) -> f(x, s(x))
   f(x, s(y)) -> f(y, x)
   f(x, y) -> ack(x, y)
   ack(s(x), y) -> f(x, x)

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

   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Innermost TRS:
Rules:
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
f(s(x), y) -> f(x, s(x))
f(x, s(y)) -> f(y, x)
f(x, y) -> ack(x, y)
ack(s(x), y) -> f(x, x)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Types:
ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
0' :: 0':s:cons_ack:cons_f
s :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encArg :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
cons_ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
cons_f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_0 :: 0':s:cons_ack:cons_f
encode_s :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
hole_0':s:cons_ack:cons_f1_3 :: 0':s:cons_ack:cons_f
gen_0':s:cons_ack:cons_f2_3 :: Nat -> 0':s:cons_ack:cons_f

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

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

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(10)
Obligation:
Innermost TRS:
Rules:
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
f(s(x), y) -> f(x, s(x))
f(x, s(y)) -> f(y, x)
f(x, y) -> ack(x, y)
ack(s(x), y) -> f(x, x)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Types:
ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
0' :: 0':s:cons_ack:cons_f
s :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encArg :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
cons_ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
cons_f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_0 :: 0':s:cons_ack:cons_f
encode_s :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
hole_0':s:cons_ack:cons_f1_3 :: 0':s:cons_ack:cons_f
gen_0':s:cons_ack:cons_f2_3 :: Nat -> 0':s:cons_ack:cons_f


Generator Equations:
gen_0':s:cons_ack:cons_f2_3(0) <=> 0'
gen_0':s:cons_ack:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_ack:cons_f2_3(x))


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

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

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ack(gen_0':s:cons_ack:cons_f2_3(1), gen_0':s:cons_ack:cons_f2_3(+(1, n4125_3))) -> *3_3, rt in Omega(n4125_3)

Induction Base:
ack(gen_0':s:cons_ack:cons_f2_3(1), gen_0':s:cons_ack:cons_f2_3(+(1, 0)))

Induction Step:
ack(gen_0':s:cons_ack:cons_f2_3(1), gen_0':s:cons_ack:cons_f2_3(+(1, +(n4125_3, 1)))) ->_R^Omega(1)
ack(gen_0':s:cons_ack:cons_f2_3(0), ack(s(gen_0':s:cons_ack:cons_f2_3(0)), gen_0':s:cons_ack:cons_f2_3(+(1, n4125_3)))) ->_IH
ack(gen_0':s:cons_ack:cons_f2_3(0), *3_3)

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:
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
f(s(x), y) -> f(x, s(x))
f(x, s(y)) -> f(y, x)
f(x, y) -> ack(x, y)
ack(s(x), y) -> f(x, x)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Types:
ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
0' :: 0':s:cons_ack:cons_f
s :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encArg :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
cons_ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
cons_f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_0 :: 0':s:cons_ack:cons_f
encode_s :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
hole_0':s:cons_ack:cons_f1_3 :: 0':s:cons_ack:cons_f
gen_0':s:cons_ack:cons_f2_3 :: Nat -> 0':s:cons_ack:cons_f


Generator Equations:
gen_0':s:cons_ack:cons_f2_3(0) <=> 0'
gen_0':s:cons_ack:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_ack:cons_f2_3(x))


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

They will be analysed ascendingly in the following order:
ack = f
ack < encArg
f < 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:
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
f(s(x), y) -> f(x, s(x))
f(x, s(y)) -> f(y, x)
f(x, y) -> ack(x, y)
ack(s(x), y) -> f(x, x)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Types:
ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
0' :: 0':s:cons_ack:cons_f
s :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encArg :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
cons_ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
cons_f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_ack :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_0 :: 0':s:cons_ack:cons_f
encode_s :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
encode_f :: 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f -> 0':s:cons_ack:cons_f
hole_0':s:cons_ack:cons_f1_3 :: 0':s:cons_ack:cons_f
gen_0':s:cons_ack:cons_f2_3 :: Nat -> 0':s:cons_ack:cons_f


Lemmas:
ack(gen_0':s:cons_ack:cons_f2_3(1), gen_0':s:cons_ack:cons_f2_3(+(1, n4125_3))) -> *3_3, rt in Omega(n4125_3)


Generator Equations:
gen_0':s:cons_ack:cons_f2_3(0) <=> 0'
gen_0':s:cons_ack:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_ack:cons_f2_3(x))


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

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

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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:cons_ack:cons_f2_3(n16000_3)) -> gen_0':s:cons_ack:cons_f2_3(n16000_3), rt in Omega(0)

Induction Base:
encArg(gen_0':s:cons_ack:cons_f2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:cons_ack:cons_f2_3(+(n16000_3, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:cons_ack:cons_f2_3(n16000_3))) ->_IH
s(gen_0':s:cons_ack:cons_f2_3(c16001_3))

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)