WORST_CASE(NON_POLY,?)
proof of input_jqbrRgNapb.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 10 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [FINISHED, 1407 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(g(x)) -> f(a(g(g(f(x))), g(f(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(g(z0)) -> f(a(g(g(f(z0))), g(f(z0))))
Tuples:
   F(g(z0)) -> c(F(a(g(g(f(z0))), g(f(z0)))), F(z0))
   F(g(z0)) -> c1(F(a(g(g(f(z0))), g(f(z0)))), F(z0))
S tuples:
   F(g(z0)) -> c(F(a(g(g(f(z0))), g(f(z0)))), F(z0))
   F(g(z0)) -> c1(F(a(g(g(f(z0))), g(f(z0)))), F(z0))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c1_2


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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(g(z0)) -> c(F(a(g(g(f(z0))), g(f(z0)))), F(z0))
   F(g(z0)) -> c1(F(a(g(g(f(z0))), g(f(z0)))), F(z0))

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

   f(g(z0)) -> f(a(g(g(f(z0))), g(f(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(g(z0)) -> c(F(a(g(g(f(z0))), g(f(z0)))), F(z0))
   F(g(z0)) -> c1(F(a(g(g(f(z0))), g(f(z0)))), F(z0))

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

   f(g(z0)) -> f(a(g(g(f(z0))), g(f(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(g(z0)) -> c(F(a(g(g(f(z0))), g(f(z0)))), F(z0))
F(g(z0)) -> c1(F(a(g(g(f(z0))), g(f(z0)))), F(z0))
f(g(z0)) -> f(a(g(g(f(z0))), g(f(z0))))

Types:
F :: g:a -> c:c1
g :: g:a -> g:a
c :: c:c1 -> c:c1 -> c:c1
a :: g:a -> g:a -> g:a
f :: g:a -> g:a
c1 :: c:c1 -> c:c1 -> c:c1
hole_c:c11_2 :: c:c1
hole_g:a2_2 :: g:a
gen_c:c13_2 :: Nat -> c:c1
gen_g:a4_2 :: Nat -> g:a

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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(g(z0)) -> c(F(a(g(g(f(z0))), g(f(z0)))), F(z0))
F(g(z0)) -> c1(F(a(g(g(f(z0))), g(f(z0)))), F(z0))
f(g(z0)) -> f(a(g(g(f(z0))), g(f(z0))))

Types:
F :: g:a -> c:c1
g :: g:a -> g:a
c :: c:c1 -> c:c1 -> c:c1
a :: g:a -> g:a -> g:a
f :: g:a -> g:a
c1 :: c:c1 -> c:c1 -> c:c1
hole_c:c11_2 :: c:c1
hole_g:a2_2 :: g:a
gen_c:c13_2 :: Nat -> c:c1
gen_g:a4_2 :: Nat -> g:a


Generator Equations:
gen_c:c13_2(0) <=> hole_c:c11_2
gen_c:c13_2(+(x, 1)) <=> c(hole_c:c11_2, gen_c:c13_2(x))
gen_g:a4_2(0) <=> hole_g:a2_2
gen_g:a4_2(+(x, 1)) <=> g(gen_g:a4_2(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_g:a4_2(+(1, n6_2))) -> *5_2, rt in Omega(EXP)

Induction Base:
f(gen_g:a4_2(+(1, 0)))

Induction Step:
f(gen_g:a4_2(+(1, +(n6_2, 1)))) ->_R^Omega(0)
f(a(g(g(f(gen_g:a4_2(+(1, n6_2))))), g(f(gen_g:a4_2(+(1, n6_2)))))) ->_IH
f(a(g(g(*5_2)), g(f(gen_g:a4_2(+(1, n6_2)))))) ->_IH
f(a(g(g(*5_2)), g(*5_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)