WORST_CASE(NON_POLY,?)
proof of input_scGp4nafEr.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 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), 14 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [FINISHED, 440 ms]
(12) BOUNDS(EXP, INF)


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(0)
Obligation:
The Runtime Complexity (parallel-innermost) 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: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0)))
   F(0) -> c1
   P(s(z0)) -> c2
S tuples:
   F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0)))
   F(0) -> c1
   P(s(z0)) -> c2
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1, P_1

Compound Symbols:   c_3, c1, c2


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(3) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF).


The TRS R consists of the following rules:

   F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0)))
   F(0) -> c1
   P(s(z0)) -> c2

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

   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0

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(EXP, INF).


The TRS R consists of the following rules:

   F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0)))
   F(0') -> c1
   P(s(z0)) -> c2

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

   f(s(z0)) -> s(f(f(p(s(z0)))))
   f(0') -> 0'
   p(s(z0)) -> z0

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(8)
Obligation:
Innermost TRS:
Rules:
F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0)))
F(0') -> c1
P(s(z0)) -> c2
f(s(z0)) -> s(f(f(p(s(z0)))))
f(0') -> 0'
p(s(z0)) -> z0

Types:
F :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c:c1 -> c:c1 -> c2 -> c:c1
f :: s:0' -> s:0'
p :: s:0' -> s:0'
P :: s:0' -> c2
0' :: s:0'
c1 :: c:c1
c2 :: c2
hole_c:c11_3 :: c:c1
hole_s:0'2_3 :: s:0'
hole_c23_3 :: c2
gen_c:c14_3 :: Nat -> c:c1
gen_s:0'5_3 :: Nat -> s:0'

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

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

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(10)
Obligation:
Innermost TRS:
Rules:
F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0)))
F(0') -> c1
P(s(z0)) -> c2
f(s(z0)) -> s(f(f(p(s(z0)))))
f(0') -> 0'
p(s(z0)) -> z0

Types:
F :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c:c1 -> c:c1 -> c2 -> c:c1
f :: s:0' -> s:0'
p :: s:0' -> s:0'
P :: s:0' -> c2
0' :: s:0'
c1 :: c:c1
c2 :: c2
hole_c:c11_3 :: c:c1
hole_s:0'2_3 :: s:0'
hole_c23_3 :: c2
gen_c:c14_3 :: Nat -> c:c1
gen_s:0'5_3 :: Nat -> s:0'


Generator Equations:
gen_c:c14_3(0) <=> c1
gen_c:c14_3(+(x, 1)) <=> c(c1, gen_c:c14_3(x), c2)
gen_s:0'5_3(0) <=> 0'
gen_s:0'5_3(+(x, 1)) <=> s(gen_s:0'5_3(x))


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

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

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(11) RewriteLemmaProof (FINISHED)
Proved the following rewrite lemma:
f(gen_s:0'5_3(n7_3)) -> gen_s:0'5_3(n7_3), rt in Omega(EXP)

Induction Base:
f(gen_s:0'5_3(0)) ->_R^Omega(0)
0'

Induction Step:
f(gen_s:0'5_3(+(n7_3, 1))) ->_R^Omega(0)
s(f(f(p(s(gen_s:0'5_3(n7_3)))))) ->_R^Omega(0)
s(f(f(gen_s:0'5_3(n7_3)))) ->_IH
s(f(gen_s:0'5_3(c8_3))) ->_IH
s(gen_s:0'5_3(c8_3))

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)