WORST_CASE(NON_POLY,?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 231 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 2 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [FINISHED, 1542 ms]
(10) 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:

   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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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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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:

   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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(3) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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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) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
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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(7) 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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(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


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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(9) 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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(10)
BOUNDS(EXP, INF)