WORST_CASE(NON_POLY,?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 2 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [FINISHED, 14.4 s]
(8) BOUNDS(EXP, INF)


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(0)
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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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(2)
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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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
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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(5) 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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(6)
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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(7) 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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(8)
BOUNDS(EXP, INF)