WORST_CASE(NON_POLY,?)
proof of input_hx63apAgDE.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 12 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [FINISHED, 14.3 s]
(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:

   g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(y))))

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:
   g(f(z0, z1)) -> f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))
Tuples:
   G(f(z0, z1)) -> c(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c1(G(g(z1)), G(z1))
   G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
S tuples:
   G(f(z0, z1)) -> c(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c1(G(g(z1)), G(z1))
   G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   G_1

Compound Symbols:   c_2, c1_2, c2_2, c3_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:

   G(f(z0, z1)) -> c(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c1(G(g(z1)), G(z1))
   G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))

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

   g(f(z0, z1)) -> f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))

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:

   G(f(z0, z1)) -> c(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c1(G(g(z1)), G(z1))
   G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))

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

   g(f(z0, z1)) -> f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(8)
Obligation:
Innermost TRS:
Rules:
G(f(z0, z1)) -> c(G(g(z0)), G(z0))
G(f(z0, z1)) -> c1(G(g(z1)), G(z1))
G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
g(f(z0, z1)) -> f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))

Types:
G :: f -> c:c1:c2:c3
f :: f -> f -> f
c :: c:c1:c2:c3 -> c:c1:c2:c3 -> c:c1:c2:c3
g :: f -> f
c1 :: c:c1:c2:c3 -> c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3 -> c:c1:c2:c3
hole_c:c1:c2:c31_4 :: c:c1:c2:c3
hole_f2_4 :: f
gen_c:c1:c2:c33_4 :: Nat -> c:c1:c2:c3
gen_f4_4 :: Nat -> f

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

They will be analysed ascendingly in the following order:
g < G

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(10)
Obligation:
Innermost TRS:
Rules:
G(f(z0, z1)) -> c(G(g(z0)), G(z0))
G(f(z0, z1)) -> c1(G(g(z1)), G(z1))
G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
g(f(z0, z1)) -> f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))

Types:
G :: f -> c:c1:c2:c3
f :: f -> f -> f
c :: c:c1:c2:c3 -> c:c1:c2:c3 -> c:c1:c2:c3
g :: f -> f
c1 :: c:c1:c2:c3 -> c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3 -> c:c1:c2:c3
hole_c:c1:c2:c31_4 :: c:c1:c2:c3
hole_f2_4 :: f
gen_c:c1:c2:c33_4 :: Nat -> c:c1:c2:c3
gen_f4_4 :: Nat -> f


Generator Equations:
gen_c:c1:c2:c33_4(0) <=> hole_c:c1:c2:c31_4
gen_c:c1:c2:c33_4(+(x, 1)) <=> c(hole_c:c1:c2:c31_4, gen_c:c1:c2:c33_4(x))
gen_f4_4(0) <=> hole_f2_4
gen_f4_4(+(x, 1)) <=> f(hole_f2_4, gen_f4_4(x))


The following defined symbols remain to be analysed:
g, G

They will be analysed ascendingly in the following order:
g < G

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(11) RewriteLemmaProof (FINISHED)
Proved the following rewrite lemma:
g(gen_f4_4(+(1, n6_4))) -> *5_4, rt in Omega(EXP)

Induction Base:
g(gen_f4_4(+(1, 0)))

Induction Step:
g(gen_f4_4(+(1, +(n6_4, 1)))) ->_R^Omega(0)
f(f(g(g(hole_f2_4)), g(g(gen_f4_4(+(1, n6_4))))), f(g(g(hole_f2_4)), g(g(gen_f4_4(+(1, n6_4)))))) ->_IH
f(f(g(g(hole_f2_4)), g(*5_4)), f(g(g(hole_f2_4)), g(g(gen_f4_4(+(1, n6_4)))))) ->_IH
f(f(g(g(hole_f2_4)), g(*5_4)), f(g(g(hole_f2_4)), g(*5_4)))

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)