WORST_CASE(Omega(n^1),?)
proof of input_Qr5Xy4OqeJ.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, 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 [LOWER BOUND(ID), 359 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 46 ms]
        (18) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs)
   lookup(Nil, Cons(x, xs)) -> x
   run(e, p) -> intlookup(e, p)
   intlookup(e, p) -> intlookup(lookup(e, p), p)

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:
   lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
   lookup(Nil, Cons(z0, z1)) -> z0
   run(z0, z1) -> intlookup(z0, z1)
   intlookup(z0, z1) -> intlookup(lookup(z0, z1), z1)
Tuples:
   LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3))
   LOOKUP(Nil, Cons(z0, z1)) -> c1
   RUN(z0, z1) -> c2(INTLOOKUP(z0, z1))
   INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
S tuples:
   LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3))
   LOOKUP(Nil, Cons(z0, z1)) -> c1
   RUN(z0, z1) -> c2(INTLOOKUP(z0, z1))
   INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
K tuples:none
Defined Rule Symbols:   lookup_2, run_2, intlookup_2

Defined Pair Symbols:   LOOKUP_2, RUN_2, INTLOOKUP_2

Compound Symbols:   c_1, c1, c2_1, 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(n^1, INF).


The TRS R consists of the following rules:

   LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3))
   LOOKUP(Nil, Cons(z0, z1)) -> c1
   RUN(z0, z1) -> c2(INTLOOKUP(z0, z1))
   INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))

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

   lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
   lookup(Nil, Cons(z0, z1)) -> z0
   run(z0, z1) -> intlookup(z0, z1)
   intlookup(z0, z1) -> intlookup(lookup(z0, z1), 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(n^1, INF).


The TRS R consists of the following rules:

   LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3))
   LOOKUP(Nil, Cons(z0, z1)) -> c1
   RUN(z0, z1) -> c2(INTLOOKUP(z0, z1))
   INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))

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

   lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
   lookup(Nil, Cons(z0, z1)) -> z0
   run(z0, z1) -> intlookup(z0, z1)
   intlookup(z0, z1) -> intlookup(lookup(z0, z1), 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:
LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3))
LOOKUP(Nil, Cons(z0, z1)) -> c1
RUN(z0, z1) -> c2(INTLOOKUP(z0, z1))
INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
lookup(Nil, Cons(z0, z1)) -> z0
run(z0, z1) -> intlookup(z0, z1)
intlookup(z0, z1) -> intlookup(lookup(z0, z1), z1)

Types:
LOOKUP :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
RUN :: Cons:Nil -> Cons:Nil -> c2
c2 :: c3 -> c2
INTLOOKUP :: Cons:Nil -> Cons:Nil -> c3
c3 :: c3 -> c:c1 -> c3
lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil
run :: Cons:Nil -> Cons:Nil -> run:intlookup
intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup
hole_c:c11_4 :: c:c1
hole_Cons:Nil2_4 :: Cons:Nil
hole_c23_4 :: c2
hole_c34_4 :: c3
hole_run:intlookup5_4 :: run:intlookup
gen_c:c16_4 :: Nat -> c:c1
gen_Cons:Nil7_4 :: Nat -> Cons:Nil
gen_c38_4 :: Nat -> c3

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LOOKUP, INTLOOKUP, lookup, intlookup

They will be analysed ascendingly in the following order:
LOOKUP < INTLOOKUP
lookup < INTLOOKUP
lookup < intlookup

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(10)
Obligation:
Innermost TRS:
Rules:
LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3))
LOOKUP(Nil, Cons(z0, z1)) -> c1
RUN(z0, z1) -> c2(INTLOOKUP(z0, z1))
INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
lookup(Nil, Cons(z0, z1)) -> z0
run(z0, z1) -> intlookup(z0, z1)
intlookup(z0, z1) -> intlookup(lookup(z0, z1), z1)

Types:
LOOKUP :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
RUN :: Cons:Nil -> Cons:Nil -> c2
c2 :: c3 -> c2
INTLOOKUP :: Cons:Nil -> Cons:Nil -> c3
c3 :: c3 -> c:c1 -> c3
lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil
run :: Cons:Nil -> Cons:Nil -> run:intlookup
intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup
hole_c:c11_4 :: c:c1
hole_Cons:Nil2_4 :: Cons:Nil
hole_c23_4 :: c2
hole_c34_4 :: c3
hole_run:intlookup5_4 :: run:intlookup
gen_c:c16_4 :: Nat -> c:c1
gen_Cons:Nil7_4 :: Nat -> Cons:Nil
gen_c38_4 :: Nat -> c3


Generator Equations:
gen_c:c16_4(0) <=> c1
gen_c:c16_4(+(x, 1)) <=> c(gen_c:c16_4(x))
gen_Cons:Nil7_4(0) <=> Nil
gen_Cons:Nil7_4(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil7_4(x))
gen_c38_4(0) <=> hole_c34_4
gen_c38_4(+(x, 1)) <=> c3(gen_c38_4(x), c1)


The following defined symbols remain to be analysed:
LOOKUP, INTLOOKUP, lookup, intlookup

They will be analysed ascendingly in the following order:
LOOKUP < INTLOOKUP
lookup < INTLOOKUP
lookup < intlookup

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LOOKUP(gen_Cons:Nil7_4(n10_4), gen_Cons:Nil7_4(+(1, n10_4))) -> gen_c:c16_4(n10_4), rt in Omega(1 + n10_4)

Induction Base:
LOOKUP(gen_Cons:Nil7_4(0), gen_Cons:Nil7_4(+(1, 0))) ->_R^Omega(1)
c1

Induction Step:
LOOKUP(gen_Cons:Nil7_4(+(n10_4, 1)), gen_Cons:Nil7_4(+(1, +(n10_4, 1)))) ->_R^Omega(1)
c(LOOKUP(gen_Cons:Nil7_4(n10_4), gen_Cons:Nil7_4(+(1, n10_4)))) ->_IH
c(gen_c:c16_4(c11_4))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(12)
Complex Obligation (BEST)

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(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3))
LOOKUP(Nil, Cons(z0, z1)) -> c1
RUN(z0, z1) -> c2(INTLOOKUP(z0, z1))
INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
lookup(Nil, Cons(z0, z1)) -> z0
run(z0, z1) -> intlookup(z0, z1)
intlookup(z0, z1) -> intlookup(lookup(z0, z1), z1)

Types:
LOOKUP :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
RUN :: Cons:Nil -> Cons:Nil -> c2
c2 :: c3 -> c2
INTLOOKUP :: Cons:Nil -> Cons:Nil -> c3
c3 :: c3 -> c:c1 -> c3
lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil
run :: Cons:Nil -> Cons:Nil -> run:intlookup
intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup
hole_c:c11_4 :: c:c1
hole_Cons:Nil2_4 :: Cons:Nil
hole_c23_4 :: c2
hole_c34_4 :: c3
hole_run:intlookup5_4 :: run:intlookup
gen_c:c16_4 :: Nat -> c:c1
gen_Cons:Nil7_4 :: Nat -> Cons:Nil
gen_c38_4 :: Nat -> c3


Generator Equations:
gen_c:c16_4(0) <=> c1
gen_c:c16_4(+(x, 1)) <=> c(gen_c:c16_4(x))
gen_Cons:Nil7_4(0) <=> Nil
gen_Cons:Nil7_4(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil7_4(x))
gen_c38_4(0) <=> hole_c34_4
gen_c38_4(+(x, 1)) <=> c3(gen_c38_4(x), c1)


The following defined symbols remain to be analysed:
LOOKUP, INTLOOKUP, lookup, intlookup

They will be analysed ascendingly in the following order:
LOOKUP < INTLOOKUP
lookup < INTLOOKUP
lookup < intlookup

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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Innermost TRS:
Rules:
LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3))
LOOKUP(Nil, Cons(z0, z1)) -> c1
RUN(z0, z1) -> c2(INTLOOKUP(z0, z1))
INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
lookup(Nil, Cons(z0, z1)) -> z0
run(z0, z1) -> intlookup(z0, z1)
intlookup(z0, z1) -> intlookup(lookup(z0, z1), z1)

Types:
LOOKUP :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
RUN :: Cons:Nil -> Cons:Nil -> c2
c2 :: c3 -> c2
INTLOOKUP :: Cons:Nil -> Cons:Nil -> c3
c3 :: c3 -> c:c1 -> c3
lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil
run :: Cons:Nil -> Cons:Nil -> run:intlookup
intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup
hole_c:c11_4 :: c:c1
hole_Cons:Nil2_4 :: Cons:Nil
hole_c23_4 :: c2
hole_c34_4 :: c3
hole_run:intlookup5_4 :: run:intlookup
gen_c:c16_4 :: Nat -> c:c1
gen_Cons:Nil7_4 :: Nat -> Cons:Nil
gen_c38_4 :: Nat -> c3


Lemmas:
LOOKUP(gen_Cons:Nil7_4(n10_4), gen_Cons:Nil7_4(+(1, n10_4))) -> gen_c:c16_4(n10_4), rt in Omega(1 + n10_4)


Generator Equations:
gen_c:c16_4(0) <=> c1
gen_c:c16_4(+(x, 1)) <=> c(gen_c:c16_4(x))
gen_Cons:Nil7_4(0) <=> Nil
gen_Cons:Nil7_4(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil7_4(x))
gen_c38_4(0) <=> hole_c34_4
gen_c38_4(+(x, 1)) <=> c3(gen_c38_4(x), c1)


The following defined symbols remain to be analysed:
lookup, INTLOOKUP, intlookup

They will be analysed ascendingly in the following order:
lookup < INTLOOKUP
lookup < intlookup

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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lookup(gen_Cons:Nil7_4(n325_4), gen_Cons:Nil7_4(+(1, n325_4))) -> gen_Cons:Nil7_4(0), rt in Omega(0)

Induction Base:
lookup(gen_Cons:Nil7_4(0), gen_Cons:Nil7_4(+(1, 0))) ->_R^Omega(0)
Nil

Induction Step:
lookup(gen_Cons:Nil7_4(+(n325_4, 1)), gen_Cons:Nil7_4(+(1, +(n325_4, 1)))) ->_R^Omega(0)
lookup(gen_Cons:Nil7_4(n325_4), gen_Cons:Nil7_4(+(1, n325_4))) ->_IH
gen_Cons:Nil7_4(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(18)
Obligation:
Innermost TRS:
Rules:
LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3))
LOOKUP(Nil, Cons(z0, z1)) -> c1
RUN(z0, z1) -> c2(INTLOOKUP(z0, z1))
INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
lookup(Nil, Cons(z0, z1)) -> z0
run(z0, z1) -> intlookup(z0, z1)
intlookup(z0, z1) -> intlookup(lookup(z0, z1), z1)

Types:
LOOKUP :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
RUN :: Cons:Nil -> Cons:Nil -> c2
c2 :: c3 -> c2
INTLOOKUP :: Cons:Nil -> Cons:Nil -> c3
c3 :: c3 -> c:c1 -> c3
lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil
run :: Cons:Nil -> Cons:Nil -> run:intlookup
intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup
hole_c:c11_4 :: c:c1
hole_Cons:Nil2_4 :: Cons:Nil
hole_c23_4 :: c2
hole_c34_4 :: c3
hole_run:intlookup5_4 :: run:intlookup
gen_c:c16_4 :: Nat -> c:c1
gen_Cons:Nil7_4 :: Nat -> Cons:Nil
gen_c38_4 :: Nat -> c3


Lemmas:
LOOKUP(gen_Cons:Nil7_4(n10_4), gen_Cons:Nil7_4(+(1, n10_4))) -> gen_c:c16_4(n10_4), rt in Omega(1 + n10_4)
lookup(gen_Cons:Nil7_4(n325_4), gen_Cons:Nil7_4(+(1, n325_4))) -> gen_Cons:Nil7_4(0), rt in Omega(0)


Generator Equations:
gen_c:c16_4(0) <=> c1
gen_c:c16_4(+(x, 1)) <=> c(gen_c:c16_4(x))
gen_Cons:Nil7_4(0) <=> Nil
gen_Cons:Nil7_4(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil7_4(x))
gen_c38_4(0) <=> hole_c34_4
gen_c38_4(+(x, 1)) <=> c3(gen_c38_4(x), c1)


The following defined symbols remain to be analysed:
INTLOOKUP, intlookup