MAYBE
proof of input_tP2JPMBA6L.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 352 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 100 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 660 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 249 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (38) CdtProblem
    (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(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
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(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
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(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
----------------------------------------

(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) [1]
   lookup(Nil, Cons(x, xs)) -> x [1]
   run(e, p) -> intlookup(e, p) [1]
   intlookup(e, p) -> intlookup(lookup(e, p), p) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) [1]
   lookup(Nil, Cons(x, xs)) -> x [1]
   run(e, p) -> intlookup(e, p) [1]
   intlookup(e, p) -> intlookup(lookup(e, p), p) [1]

The TRS has the following type information:
lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
run :: Cons:Nil -> Cons:Nil -> run:intlookup
intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   run_2
   intlookup_2

(c) The following functions are completely defined:

   lookup_2

Due to the following rules being added:

   lookup(v0, v1) -> Nil [0]

And the following fresh constants:    const

----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) [1]
   lookup(Nil, Cons(x, xs)) -> x [1]
   run(e, p) -> intlookup(e, p) [1]
   intlookup(e, p) -> intlookup(lookup(e, p), p) [1]
   lookup(v0, v1) -> Nil [0]

The TRS has the following type information:
lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
run :: Cons:Nil -> Cons:Nil -> run:intlookup
intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup
const :: run:intlookup

Rewrite Strategy: INNERMOST
----------------------------------------

(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) [1]
   lookup(Nil, Cons(x, xs)) -> x [1]
   run(e, p) -> intlookup(e, p) [1]
   intlookup(Cons(x'', xs''), Cons(x1, xs1)) -> intlookup(lookup(xs'', xs1), Cons(x1, xs1)) [2]
   intlookup(Nil, Cons(x2, xs2)) -> intlookup(x2, Cons(x2, xs2)) [2]
   intlookup(e, p) -> intlookup(Nil, p) [1]
   lookup(v0, v1) -> Nil [0]

The TRS has the following type information:
lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
run :: Cons:Nil -> Cons:Nil -> run:intlookup
intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup
const :: run:intlookup

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   const => 0

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0
   intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1
   intlookup(z, z') -{ 1 }-> intlookup(0, p) :|: z = e, z' = p, p >= 0, e >= 0
   lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lookup(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   run(z, z') -{ 1 }-> intlookup(e, p) :|: z = e, z' = p, p >= 0, e >= 0


----------------------------------------

(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0
   intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1
   intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0
   lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0


----------------------------------------

(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { lookup }
   { intlookup }
   { run }

----------------------------------------

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0
   intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1
   intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0
   lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0

Function symbols to be analyzed: {lookup}, {intlookup}, {run}

----------------------------------------

(19) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0
   intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1
   intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0
   lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0

Function symbols to be analyzed: {lookup}, {intlookup}, {run}

----------------------------------------

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: lookup
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0
   intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1
   intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0
   lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0

Function symbols to be analyzed: {lookup}, {intlookup}, {run}
Previous analysis results are:
  lookup: runtime: ?, size: O(n^1) [z']

----------------------------------------

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: lookup
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0
   intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1
   intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0
   lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0

Function symbols to be analyzed: {intlookup}, {run}
Previous analysis results are:
  lookup: runtime: O(n^1) [2 + z'], size: O(n^1) [z']

----------------------------------------

(25) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   intlookup(z, z') -{ 4 + xs1 }-> intlookup(s', 1 + x1 + xs1) :|: s' >= 0, s' <= xs1, x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1
   intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0
   intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0
   lookup(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= xs, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0

Function symbols to be analyzed: {intlookup}, {run}
Previous analysis results are:
  lookup: runtime: O(n^1) [2 + z'], size: O(n^1) [z']

----------------------------------------

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: intlookup
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   intlookup(z, z') -{ 4 + xs1 }-> intlookup(s', 1 + x1 + xs1) :|: s' >= 0, s' <= xs1, x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1
   intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0
   intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0
   lookup(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= xs, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0

Function symbols to be analyzed: {intlookup}, {run}
Previous analysis results are:
  lookup: runtime: O(n^1) [2 + z'], size: O(n^1) [z']
  intlookup: runtime: ?, size: O(1) [0]

----------------------------------------

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: intlookup
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   intlookup(z, z') -{ 4 + xs1 }-> intlookup(s', 1 + x1 + xs1) :|: s' >= 0, s' <= xs1, x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1
   intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0
   intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0
   lookup(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= xs, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0

Function symbols to be analyzed: {intlookup}, {run}
Previous analysis results are:
  lookup: runtime: O(n^1) [2 + z'], size: O(n^1) [z']
  intlookup: runtime: INF, size: O(1) [0]

----------------------------------------

(31) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   lookup(v0, v1) -> null_lookup [0]

And the following fresh constants:    null_lookup, const

----------------------------------------

(32)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) [1]
   lookup(Nil, Cons(x, xs)) -> x [1]
   run(e, p) -> intlookup(e, p) [1]
   intlookup(e, p) -> intlookup(lookup(e, p), p) [1]
   lookup(v0, v1) -> null_lookup [0]

The TRS has the following type information:
lookup :: Cons:Nil:null_lookup -> Cons:Nil:null_lookup -> Cons:Nil:null_lookup
Cons :: Cons:Nil:null_lookup -> Cons:Nil:null_lookup -> Cons:Nil:null_lookup
Nil :: Cons:Nil:null_lookup
run :: Cons:Nil:null_lookup -> Cons:Nil:null_lookup -> run:intlookup
intlookup :: Cons:Nil:null_lookup -> Cons:Nil:null_lookup -> run:intlookup
null_lookup :: Cons:Nil:null_lookup
const :: run:intlookup

Rewrite Strategy: INNERMOST
----------------------------------------

(33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   null_lookup => 0
   const => 0

----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   intlookup(z, z') -{ 1 }-> intlookup(lookup(e, p), p) :|: z = e, z' = p, p >= 0, e >= 0
   lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lookup(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   run(z, z') -{ 1 }-> intlookup(e, p) :|: z = e, z' = p, p >= 0, e >= 0

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(35) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(36)
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


----------------------------------------

(37) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   RUN(z0, z1) -> c2(INTLOOKUP(z0, z1))
Removed 1 trailing nodes:
   LOOKUP(Nil, Cons(z0, z1)) -> c1

----------------------------------------

(38)
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))
   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))
   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, INTLOOKUP_2

Compound Symbols:   c_1, c3_2


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(39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   run(z0, z1) -> intlookup(z0, z1)
   intlookup(z0, z1) -> intlookup(lookup(z0, z1), z1)

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(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
   lookup(Nil, Cons(z0, z1)) -> z0
Tuples:
   LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3))
   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))
   INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
K tuples:none
Defined Rule Symbols:   lookup_2

Defined Pair Symbols:   LOOKUP_2, INTLOOKUP_2

Compound Symbols:   c_1, c3_2


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(41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3)) by 
   LOOKUP(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c(LOOKUP(Cons(y0, y1), Cons(y2, y3)))

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(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
   lookup(Nil, Cons(z0, z1)) -> z0
Tuples:
   INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
   LOOKUP(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c(LOOKUP(Cons(y0, y1), Cons(y2, y3)))
S tuples:
   INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
   LOOKUP(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c(LOOKUP(Cons(y0, y1), Cons(y2, y3)))
K tuples:none
Defined Rule Symbols:   lookup_2

Defined Pair Symbols:   INTLOOKUP_2, LOOKUP_2

Compound Symbols:   c3_2, c_1


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(43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LOOKUP(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c(LOOKUP(Cons(y0, y1), Cons(y2, y3))) by 
   LOOKUP(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c(LOOKUP(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5))))

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(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3)
   lookup(Nil, Cons(z0, z1)) -> z0
Tuples:
   INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
   LOOKUP(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c(LOOKUP(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5))))
S tuples:
   INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1))
   LOOKUP(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c(LOOKUP(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5))))
K tuples:none
Defined Rule Symbols:   lookup_2

Defined Pair Symbols:   INTLOOKUP_2, LOOKUP_2

Compound Symbols:   c3_2, c_1