/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(EXP, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 64 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 [FINISHED, 478 ms]
(12) BOUNDS(EXP, INF)


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(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   f(s(x)) -> s(f(f(p(s(x)))))
   f(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: FULL
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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(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0


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(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   f(s(x)) -> s(f(f(p(s(x)))))
   f(0) -> 0
   p(s(x)) -> x

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0

Rewrite Strategy: FULL
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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 (full) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   f(s(x)) -> s(f(f(p(s(x)))))
   f(0) -> 0
   p(s(x)) -> x

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0

Rewrite Strategy: FULL
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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 (full) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   f(s(x)) -> s(f(f(p(s(x)))))
   f(0') -> 0'
   p(s(x)) -> x

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0') -> 0'
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0'

Rewrite Strategy: FULL
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
TRS:
Rules:
f(s(x)) -> s(f(f(p(s(x)))))
f(0') -> 0'
p(s(x)) -> x
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_0 -> 0'

Types:
f :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
s :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
p :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
0' :: s:0':cons_f:cons_p
encArg :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
cons_f :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
cons_p :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
encode_f :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
encode_s :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
encode_p :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
encode_0 :: s:0':cons_f:cons_p
hole_s:0':cons_f:cons_p1_2 :: s:0':cons_f:cons_p
gen_s:0':cons_f:cons_p2_2 :: Nat -> s:0':cons_f:cons_p

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

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

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(10)
Obligation:
TRS:
Rules:
f(s(x)) -> s(f(f(p(s(x)))))
f(0') -> 0'
p(s(x)) -> x
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_0 -> 0'

Types:
f :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
s :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
p :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
0' :: s:0':cons_f:cons_p
encArg :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
cons_f :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
cons_p :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
encode_f :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
encode_s :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
encode_p :: s:0':cons_f:cons_p -> s:0':cons_f:cons_p
encode_0 :: s:0':cons_f:cons_p
hole_s:0':cons_f:cons_p1_2 :: s:0':cons_f:cons_p
gen_s:0':cons_f:cons_p2_2 :: Nat -> s:0':cons_f:cons_p


Generator Equations:
gen_s:0':cons_f:cons_p2_2(0) <=> 0'
gen_s:0':cons_f:cons_p2_2(+(x, 1)) <=> s(gen_s:0':cons_f:cons_p2_2(x))


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

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

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(11) RewriteLemmaProof (FINISHED)
Proved the following rewrite lemma:
f(gen_s:0':cons_f:cons_p2_2(n4_2)) -> gen_s:0':cons_f:cons_p2_2(n4_2), rt in Omega(EXP)

Induction Base:
f(gen_s:0':cons_f:cons_p2_2(0)) ->_R^Omega(1)
0'

Induction Step:
f(gen_s:0':cons_f:cons_p2_2(+(n4_2, 1))) ->_R^Omega(1)
s(f(f(p(s(gen_s:0':cons_f:cons_p2_2(n4_2)))))) ->_R^Omega(1)
s(f(f(gen_s:0':cons_f:cons_p2_2(n4_2)))) ->_IH
s(f(gen_s:0':cons_f:cons_p2_2(c5_2))) ->_IH
s(gen_s:0':cons_f:cons_p2_2(c5_2))

We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP
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(12)
BOUNDS(EXP, INF)