MAYBE
proof of input_9P6XHdgT9t.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (12) CpxRNTS
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
        (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 1266 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 332 ms]
        (28) CpxRNTS
(29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(30) CdtProblem
    (31) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (32) CdtProblem
    (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) 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:

   g(Cons(x, xs), y) -> Cons(x, xs)
   h(Nil, y) -> h(Nil, y)
   h(Cons(x, xs), y) -> f(Cons(x, xs), y)
   g(Nil, y) -> h(Nil, y)
   f(Nil, y) -> g(Nil, y)
   f(Cons(x, xs), y) -> h(Cons(x, xs), y)
   sp1(x, y) -> f(x, y)
   r(x, y) -> x

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:

   g(Cons(x, xs), y) -> Cons(x, xs)
   h(Nil, y) -> h(Nil, y)
   h(Cons(x, xs), y) -> f(Cons(x, xs), y)
   g(Nil, y) -> h(Nil, y)
   f(Nil, y) -> g(Nil, y)
   f(Cons(x, xs), y) -> h(Cons(x, xs), y)
   sp1(x, y) -> f(x, y)
   r(x, y) -> x

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:

   g(Cons(x, xs), y) -> Cons(x, xs)
   h(Nil, y) -> h(Nil, y)
   h(Cons(x, xs), y) -> f(Cons(x, xs), y)
   g(Nil, y) -> h(Nil, y)
   f(Nil, y) -> g(Nil, y)
   f(Cons(x, xs), y) -> h(Cons(x, xs), y)
   sp1(x, y) -> f(x, y)
   r(x, y) -> x

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:

   g(Cons(x, xs), y) -> Cons(x, xs) [1]
   h(Nil, y) -> h(Nil, y) [1]
   h(Cons(x, xs), y) -> f(Cons(x, xs), y) [1]
   g(Nil, y) -> h(Nil, y) [1]
   f(Nil, y) -> g(Nil, y) [1]
   f(Cons(x, xs), y) -> h(Cons(x, xs), y) [1]
   sp1(x, y) -> f(x, y) [1]
   r(x, y) -> x [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:

   g(Cons(x, xs), y) -> Cons(x, xs) [1]
   h(Nil, y) -> h(Nil, y) [1]
   h(Cons(x, xs), y) -> f(Cons(x, xs), y) [1]
   g(Nil, y) -> h(Nil, y) [1]
   f(Nil, y) -> g(Nil, y) [1]
   f(Cons(x, xs), y) -> h(Cons(x, xs), y) [1]
   sp1(x, y) -> f(x, y) [1]
   r(x, y) -> x [1]

The TRS has the following type information:
g :: Cons:Nil -> c -> Cons:Nil
Cons :: a -> b -> Cons:Nil
h :: Cons:Nil -> c -> Cons:Nil
Nil :: Cons:Nil
f :: Cons:Nil -> c -> Cons:Nil
sp1 :: Cons:Nil -> c -> Cons:Nil
r :: r -> d -> r

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

(9) 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:
none

And the following fresh constants:    const, const1, const2, const3, const4

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

(10)
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:

   g(Cons(x, xs), y) -> Cons(x, xs) [1]
   h(Nil, y) -> h(Nil, y) [1]
   h(Cons(x, xs), y) -> f(Cons(x, xs), y) [1]
   g(Nil, y) -> h(Nil, y) [1]
   f(Nil, y) -> g(Nil, y) [1]
   f(Cons(x, xs), y) -> h(Cons(x, xs), y) [1]
   sp1(x, y) -> f(x, y) [1]
   r(x, y) -> x [1]

The TRS has the following type information:
g :: Cons:Nil -> c -> Cons:Nil
Cons :: a -> b -> Cons:Nil
h :: Cons:Nil -> c -> Cons:Nil
Nil :: Cons:Nil
f :: Cons:Nil -> c -> Cons:Nil
sp1 :: Cons:Nil -> c -> Cons:Nil
r :: r -> d -> r
const :: c
const1 :: a
const2 :: b
const3 :: r
const4 :: d

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

(11) 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
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> h(1 + x + xs, y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   f(z, z') -{ 1 }-> g(0, y) :|: y >= 0, z = 0, z' = y
   g(z, z') -{ 1 }-> h(0, y) :|: y >= 0, z = 0, z' = y
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   h(z, z') -{ 1 }-> h(0, y) :|: y >= 0, z = 0, z' = y
   h(z, z') -{ 1 }-> f(1 + x + xs, y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   r(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   sp1(z, z') -{ 1 }-> f(x, y) :|: x >= 0, y >= 0, z = x, z' = y

Only complete derivations are relevant for the runtime complexity.

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

(13) 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:

   g_2
   h_2
   f_2
   sp1_2
   r_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1, const2, const3, const4

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

(14)
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:

   g(Cons(x, xs), y) -> Cons(x, xs) [1]
   h(Nil, y) -> h(Nil, y) [1]
   h(Cons(x, xs), y) -> f(Cons(x, xs), y) [1]
   g(Nil, y) -> h(Nil, y) [1]
   f(Nil, y) -> g(Nil, y) [1]
   f(Cons(x, xs), y) -> h(Cons(x, xs), y) [1]
   sp1(x, y) -> f(x, y) [1]
   r(x, y) -> x [1]

The TRS has the following type information:
g :: Cons:Nil -> c -> Cons:Nil
Cons :: a -> b -> Cons:Nil
h :: Cons:Nil -> c -> Cons:Nil
Nil :: Cons:Nil
f :: Cons:Nil -> c -> Cons:Nil
sp1 :: Cons:Nil -> c -> Cons:Nil
r :: r -> d -> r
const :: c
const1 :: a
const2 :: b
const3 :: r
const4 :: d

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

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

(16)
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:

   g(Cons(x, xs), y) -> Cons(x, xs) [1]
   h(Nil, y) -> h(Nil, y) [1]
   h(Cons(x, xs), y) -> f(Cons(x, xs), y) [1]
   g(Nil, y) -> h(Nil, y) [1]
   f(Nil, y) -> g(Nil, y) [1]
   f(Cons(x, xs), y) -> h(Cons(x, xs), y) [1]
   sp1(x, y) -> f(x, y) [1]
   r(x, y) -> x [1]

The TRS has the following type information:
g :: Cons:Nil -> c -> Cons:Nil
Cons :: a -> b -> Cons:Nil
h :: Cons:Nil -> c -> Cons:Nil
Nil :: Cons:Nil
f :: Cons:Nil -> c -> Cons:Nil
sp1 :: Cons:Nil -> c -> Cons:Nil
r :: r -> d -> r
const :: c
const1 :: a
const2 :: b
const3 :: r
const4 :: d

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

(17) 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
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0

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

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

   f(z, z') -{ 1 }-> h(1 + x + xs, y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   f(z, z') -{ 1 }-> g(0, y) :|: y >= 0, z = 0, z' = y
   g(z, z') -{ 1 }-> h(0, y) :|: y >= 0, z = 0, z' = y
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   h(z, z') -{ 1 }-> h(0, y) :|: y >= 0, z = 0, z' = y
   h(z, z') -{ 1 }-> f(1 + x + xs, y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   r(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   sp1(z, z') -{ 1 }-> f(x, y) :|: x >= 0, y >= 0, z = x, z' = y


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

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

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

   f(z, z') -{ 1 }-> h(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   f(z, z') -{ 1 }-> g(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   h(z, z') -{ 1 }-> f(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   r(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   sp1(z, z') -{ 1 }-> f(z, z') :|: z >= 0, z' >= 0


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

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

   { f, h, g }
   { r }
   { sp1 }

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

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

   f(z, z') -{ 1 }-> h(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   f(z, z') -{ 1 }-> g(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   h(z, z') -{ 1 }-> f(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   r(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   sp1(z, z') -{ 1 }-> f(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,h,g}, {r}, {sp1}

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

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

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

   f(z, z') -{ 1 }-> h(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   f(z, z') -{ 1 }-> g(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   h(z, z') -{ 1 }-> f(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   r(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   sp1(z, z') -{ 1 }-> f(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,h,g}, {r}, {sp1}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

   f(z, z') -{ 1 }-> h(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   f(z, z') -{ 1 }-> g(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   h(z, z') -{ 1 }-> f(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   r(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   sp1(z, z') -{ 1 }-> f(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,h,g}, {r}, {sp1}
Previous analysis results are:
  f: runtime: ?, size: O(1) [0]
  h: runtime: ?, size: O(1) [0]
  g: runtime: ?, size: O(n^1) [z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 1 }-> h(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   f(z, z') -{ 1 }-> g(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   g(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z') :|: z' >= 0, z = 0
   h(z, z') -{ 1 }-> f(1 + x + xs, z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   r(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   sp1(z, z') -{ 1 }-> f(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,h,g}, {r}, {sp1}
Previous analysis results are:
  f: runtime: INF, size: O(1) [0]
  h: runtime: ?, size: O(1) [0]
  g: runtime: ?, size: O(n^1) [z]

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

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(Cons(z0, z1), z2) -> Cons(z0, z1)
   g(Nil, z0) -> h(Nil, z0)
   h(Nil, z0) -> h(Nil, z0)
   h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
   f(Nil, z0) -> g(Nil, z0)
   f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
   sp1(z0, z1) -> f(z0, z1)
   r(z0, z1) -> z0
Tuples:
   G(Cons(z0, z1), z2) -> c
   G(Nil, z0) -> c1(H(Nil, z0))
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Nil, z0) -> c4(G(Nil, z0))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
   SP1(z0, z1) -> c6(F(z0, z1))
   R(z0, z1) -> c7
S tuples:
   G(Cons(z0, z1), z2) -> c
   G(Nil, z0) -> c1(H(Nil, z0))
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Nil, z0) -> c4(G(Nil, z0))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
   SP1(z0, z1) -> c6(F(z0, z1))
   R(z0, z1) -> c7
K tuples:none
Defined Rule Symbols:   g_2, h_2, f_2, sp1_2, r_2

Defined Pair Symbols:   G_2, H_2, F_2, SP1_2, R_2

Compound Symbols:   c, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7


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

(31) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   SP1(z0, z1) -> c6(F(z0, z1))
   F(Nil, z0) -> c4(G(Nil, z0))
   G(Nil, z0) -> c1(H(Nil, z0))
Removed 2 trailing nodes:
   G(Cons(z0, z1), z2) -> c
   R(z0, z1) -> c7

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(Cons(z0, z1), z2) -> Cons(z0, z1)
   g(Nil, z0) -> h(Nil, z0)
   h(Nil, z0) -> h(Nil, z0)
   h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
   f(Nil, z0) -> g(Nil, z0)
   f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
   sp1(z0, z1) -> f(z0, z1)
   r(z0, z1) -> z0
Tuples:
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
S tuples:
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
K tuples:none
Defined Rule Symbols:   g_2, h_2, f_2, sp1_2, r_2

Defined Pair Symbols:   H_2, F_2

Compound Symbols:   c2_1, c3_1, c5_1


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

(33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   g(Cons(z0, z1), z2) -> Cons(z0, z1)
   g(Nil, z0) -> h(Nil, z0)
   h(Nil, z0) -> h(Nil, z0)
   h(Cons(z0, z1), z2) -> f(Cons(z0, z1), z2)
   f(Nil, z0) -> g(Nil, z0)
   f(Cons(z0, z1), z2) -> h(Cons(z0, z1), z2)
   sp1(z0, z1) -> f(z0, z1)
   r(z0, z1) -> z0

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
S tuples:
   H(Nil, z0) -> c2(H(Nil, z0))
   H(Cons(z0, z1), z2) -> c3(F(Cons(z0, z1), z2))
   F(Cons(z0, z1), z2) -> c5(H(Cons(z0, z1), z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   H_2, F_2

Compound Symbols:   c2_1, c3_1, c5_1