MAYBE
proof of input_UoNUC5qTdL.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 92 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 45 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 156 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 65 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 141 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
        (42) CpxRNTS
    (43) CompletionProof [UPPER BOUND(ID), 0 ms]
    (44) CpxTypedWeightedCompleteTrs
        (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
(47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(48) CdtProblem
    (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) 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:

   eq -> true
   eq -> eq
   eq -> false
   inf(X) -> cons
   take(0, X) -> nil
   take(s, cons) -> cons
   length(nil) -> 0
   length(cons) -> s

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:

   eq -> true
   eq -> eq
   eq -> false
   inf(X) -> cons
   take(0', X) -> nil
   take(s, cons) -> cons
   length(nil) -> 0'
   length(cons) -> s

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:

   eq -> true
   eq -> eq
   eq -> false
   inf(X) -> cons
   take(0, X) -> nil
   take(s, cons) -> cons
   length(nil) -> 0
   length(cons) -> s

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:

   eq -> true [1]
   eq -> eq [1]
   eq -> false [1]
   inf(X) -> cons [1]
   take(0, X) -> nil [1]
   take(s, cons) -> cons [1]
   length(nil) -> 0 [1]
   length(cons) -> s [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:

   eq -> true [1]
   eq -> eq [1]
   eq -> false [1]
   inf(X) -> cons [1]
   take(0, X) -> nil [1]
   take(s, cons) -> cons [1]
   length(nil) -> 0 [1]
   length(cons) -> s [1]

The TRS has the following type information:
eq :: true:false
true :: true:false
false :: true:false
inf :: a -> cons:nil
cons :: cons:nil
take :: 0:s -> cons:nil -> cons:nil
0 :: 0:s
nil :: cons:nil
s :: 0:s
length :: cons:nil -> 0:s

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:

   eq
   inf_1
   take_2
   length_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

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:

   eq -> true [1]
   eq -> eq [1]
   eq -> false [1]
   inf(X) -> cons [1]
   take(0, X) -> nil [1]
   take(s, cons) -> cons [1]
   length(nil) -> 0 [1]
   length(cons) -> s [1]

The TRS has the following type information:
eq :: true:false
true :: true:false
false :: true:false
inf :: a -> cons:nil
cons :: cons:nil
take :: 0:s -> cons:nil -> cons:nil
0 :: 0:s
nil :: cons:nil
s :: 0:s
length :: cons:nil -> 0:s
const :: a

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:

   eq -> true [1]
   eq -> eq [1]
   eq -> false [1]
   inf(X) -> cons [1]
   take(0, X) -> nil [1]
   take(s, cons) -> cons [1]
   length(nil) -> 0 [1]
   length(cons) -> s [1]

The TRS has the following type information:
eq :: true:false
true :: true:false
false :: true:false
inf :: a -> cons:nil
cons :: cons:nil
take :: 0:s -> cons:nil -> cons:nil
0 :: 0:s
nil :: cons:nil
s :: 0:s
length :: cons:nil -> 0:s
const :: a

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:

   true => 1
   false => 0
   cons => 0
   0 => 0
   nil => 1
   s => 1
   const => 0

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

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

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: X >= 0, z = X
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' = X, X >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 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:

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0


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

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

   { inf }
   { take }
   { length }
   { eq }

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

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

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {inf}, {take}, {length}, {eq}

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

(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:

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {inf}, {take}, {length}, {eq}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {inf}, {take}, {length}, {eq}
Previous analysis results are:
  inf: runtime: ?, size: O(1) [0]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {take}, {length}, {eq}
Previous analysis results are:
  inf: runtime: O(1) [1], size: O(1) [0]

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

(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:

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {take}, {length}, {eq}
Previous analysis results are:
  inf: runtime: O(1) [1], size: O(1) [0]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {take}, {length}, {eq}
Previous analysis results are:
  inf: runtime: O(1) [1], size: O(1) [0]
  take: runtime: ?, size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {length}, {eq}
Previous analysis results are:
  inf: runtime: O(1) [1], size: O(1) [0]
  take: runtime: O(1) [1], size: O(1) [1]

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

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {length}, {eq}
Previous analysis results are:
  inf: runtime: O(1) [1], size: O(1) [0]
  take: runtime: O(1) [1], size: O(1) [1]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {length}, {eq}
Previous analysis results are:
  inf: runtime: O(1) [1], size: O(1) [0]
  take: runtime: O(1) [1], size: O(1) [1]
  length: runtime: ?, size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {eq}
Previous analysis results are:
  inf: runtime: O(1) [1], size: O(1) [0]
  take: runtime: O(1) [1], size: O(1) [1]
  length: runtime: O(1) [1], size: O(1) [1]

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

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {eq}
Previous analysis results are:
  inf: runtime: O(1) [1], size: O(1) [0]
  take: runtime: O(1) [1], size: O(1) [1]
  length: runtime: O(1) [1], size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {eq}
Previous analysis results are:
  inf: runtime: O(1) [1], size: O(1) [0]
  take: runtime: O(1) [1], size: O(1) [1]
  length: runtime: O(1) [1], size: O(1) [1]
  eq: runtime: ?, size: O(1) [1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 0 :|: z >= 0
   length(z) -{ 1 }-> 1 :|: z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0

Function symbols to be analyzed: {eq}
Previous analysis results are:
  inf: runtime: O(1) [1], size: O(1) [0]
  take: runtime: O(1) [1], size: O(1) [1]
  length: runtime: O(1) [1], size: O(1) [1]
  eq: runtime: INF, size: O(1) [1]

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

(43) 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:

   take(v0, v1) -> null_take [0]
   length(v0) -> null_length [0]

And the following fresh constants:    null_take, null_length, const

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

(44)
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:

   eq -> true [1]
   eq -> eq [1]
   eq -> false [1]
   inf(X) -> cons [1]
   take(0, X) -> nil [1]
   take(s, cons) -> cons [1]
   length(nil) -> 0 [1]
   length(cons) -> s [1]
   take(v0, v1) -> null_take [0]
   length(v0) -> null_length [0]

The TRS has the following type information:
eq :: true:false
true :: true:false
false :: true:false
inf :: a -> cons:nil:null_take
cons :: cons:nil:null_take
take :: 0:s:null_length -> cons:nil:null_take -> cons:nil:null_take
0 :: 0:s:null_length
nil :: cons:nil:null_take
s :: 0:s:null_length
length :: cons:nil:null_take -> 0:s:null_length
null_take :: cons:nil:null_take
null_length :: 0:s:null_length
const :: a

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

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

   true => 1
   false => 0
   cons => 1
   0 => 1
   nil => 2
   s => 2
   null_take => 0
   null_length => 0
   const => 0

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   eq -{ 1 }-> eq :|: 
   eq -{ 1 }-> 1 :|: 
   eq -{ 1 }-> 0 :|: 
   inf(z) -{ 1 }-> 1 :|: X >= 0, z = X
   length(z) -{ 1 }-> 2 :|: z = 1
   length(z) -{ 1 }-> 1 :|: z = 2
   length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   take(z, z') -{ 1 }-> 2 :|: z' = X, z = 1, X >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1
   take(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq -> true
   eq -> eq
   eq -> false
   inf(z0) -> cons
   take(0, z0) -> nil
   take(s, cons) -> cons
   length(nil) -> 0
   length(cons) -> s
Tuples:
   EQ -> c
   EQ -> c1(EQ)
   EQ -> c2
   INF(z0) -> c3
   TAKE(0, z0) -> c4
   TAKE(s, cons) -> c5
   LENGTH(nil) -> c6
   LENGTH(cons) -> c7
S tuples:
   EQ -> c
   EQ -> c1(EQ)
   EQ -> c2
   INF(z0) -> c3
   TAKE(0, z0) -> c4
   TAKE(s, cons) -> c5
   LENGTH(nil) -> c6
   LENGTH(cons) -> c7
K tuples:none
Defined Rule Symbols:   eq, inf_1, take_2, length_1

Defined Pair Symbols:   EQ, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c, c1_1, c2, c3, c4, c5, c6, c7


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

(49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   LENGTH(nil) -> c6
   TAKE(s, cons) -> c5
   LENGTH(cons) -> c7
   TAKE(0, z0) -> c4
   EQ -> c2
   EQ -> c
   INF(z0) -> c3

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq -> true
   eq -> eq
   eq -> false
   inf(z0) -> cons
   take(0, z0) -> nil
   take(s, cons) -> cons
   length(nil) -> 0
   length(cons) -> s
Tuples:
   EQ -> c1(EQ)
S tuples:
   EQ -> c1(EQ)
K tuples:none
Defined Rule Symbols:   eq, inf_1, take_2, length_1

Defined Pair Symbols:   EQ

Compound Symbols:   c1_1


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

(51) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   eq -> true
   eq -> eq
   eq -> false
   inf(z0) -> cons
   take(0, z0) -> nil
   take(s, cons) -> cons
   length(nil) -> 0
   length(cons) -> s

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   EQ -> c1(EQ)
S tuples:
   EQ -> c1(EQ)
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   EQ

Compound Symbols:   c1_1