/export/starexec/sandbox/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [FINISHED, 455 ms]
(8) BOUNDS(EXP, INF)


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS 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) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS 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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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
TRS:
Rules:
f(s(x)) -> s(f(f(p(s(x)))))
f(0') -> 0'
p(s(x)) -> x

Types:
f :: s:0' -> s:0'
s :: s:0' -> s:0'
p :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f
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(6)
Obligation:
TRS:
Rules:
f(s(x)) -> s(f(f(p(s(x)))))
f(0') -> 0'
p(s(x)) -> x

Types:
f :: s:0' -> s:0'
s :: s:0' -> s:0'
p :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
f
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(7) RewriteLemmaProof (FINISHED)
Proved the following rewrite lemma:
f(gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(n4_0), rt in Omega(EXP)

Induction Base:
f(gen_s:0'2_0(0)) ->_R^Omega(1)
0'

Induction Step:
f(gen_s:0'2_0(+(n4_0, 1))) ->_R^Omega(1)
s(f(f(p(s(gen_s:0'2_0(n4_0)))))) ->_R^Omega(1)
s(f(f(gen_s:0'2_0(n4_0)))) ->_IH
s(f(gen_s:0'2_0(c5_0))) ->_IH
s(gen_s:0'2_0(c5_0))

We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP
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(8)
BOUNDS(EXP, INF)