MAYBE
proof of input_pshDkFS0uV.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 930 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 237 ms]
        (24) CpxRNTS
    (25) CompletionProof [UPPER BOUND(ID), 0 ms]
    (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
(29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(30) CdtProblem
    (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 54 ms]
    (36) CdtProblem
    (37) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtInstantiationProof [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
    (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtForwardInstantiationProof [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(0, 0) -> true
   eq(s(X), s(Y)) -> eq(X, Y)
   eq(X, Y) -> false
   inf(X) -> cons(X, inf(s(X)))
   take(0, X) -> nil
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   length(nil) -> 0
   length(cons(X, L)) -> s(length(L))

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

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

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

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
inf :: 0:s -> cons:nil
cons :: 0:s -> cons:nil -> cons:nil
take :: 0:s -> cons:nil -> cons:nil
nil :: cons:nil
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_2
   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: none

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

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

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

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

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

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:

   0 => 0
   true => 1
   false => 0
   nil => 0

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

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

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


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

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


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

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

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

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

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: INF with polynomial bound: ?

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

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

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

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

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

(25) 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

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

(26)
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(0, 0) -> true [1]
   eq(s(X), s(Y)) -> eq(X, Y) [1]
   eq(X, Y) -> false [1]
   inf(X) -> cons(X, inf(s(X))) [1]
   take(0, X) -> nil [1]
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1]
   length(nil) -> 0 [1]
   length(cons(X, L)) -> s(length(L)) [1]
   take(v0, v1) -> null_take [0]
   length(v0) -> null_length [0]

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

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

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

   0 => 0
   true => 1
   false => 0
   nil => 0
   null_take => 0
   null_length => 0

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(z0, z1) -> false
   inf(z0) -> cons(z0, inf(s(z0)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   EQ(0, 0) -> c
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   EQ(z0, z1) -> c2
   INF(z0) -> c3(INF(s(z0)))
   TAKE(0, z0) -> c4
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(nil) -> c6
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
S tuples:
   EQ(0, 0) -> c
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   EQ(z0, z1) -> c2
   INF(z0) -> c3(INF(s(z0)))
   TAKE(0, z0) -> c4
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(nil) -> c6
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
K tuples:none
Defined Rule Symbols:   eq_2, inf_1, take_2, length_1

Defined Pair Symbols:   EQ_2, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c, c1_1, c2, c3_1, c4, c5_1, c6, c7_1


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

(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   LENGTH(nil) -> c6
   EQ(z0, z1) -> c2
   EQ(0, 0) -> c
   TAKE(0, z0) -> c4

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(z0, z1) -> false
   inf(z0) -> cons(z0, inf(s(z0)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
S tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
K tuples:none
Defined Rule Symbols:   eq_2, inf_1, take_2, length_1

Defined Pair Symbols:   EQ_2, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c1_1, c3_1, c5_1, c7_1


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

(33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   eq(0, 0) -> true
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(z0, z1) -> false
   inf(z0) -> cons(z0, inf(s(z0)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
S tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   EQ_2, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c1_1, c3_1, c5_1, c7_1


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

(35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
We considered the (Usable) Rules:none
And the Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(EQ(x_1, x_2)) = x_1
   POL(INF(x_1)) = 0
   POL(LENGTH(x_1)) = x_1
   POL(TAKE(x_1, x_2)) = x_2
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(s(x_1)) = [1] + x_1

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
S tuples:
   INF(z0) -> c3(INF(s(z0)))
K tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   EQ_2, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c1_1, c3_1, c5_1, c7_1


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

(37) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace INF(z0) -> c3(INF(s(z0))) by 
   INF(s(x0)) -> c3(INF(s(s(x0))))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   INF(s(x0)) -> c3(INF(s(s(x0))))
S tuples:
   INF(s(x0)) -> c3(INF(s(s(x0))))
K tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   EQ_2, TAKE_2, LENGTH_1, INF_1

Compound Symbols:   c1_1, c5_1, c7_1, c3_1


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

(39) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace INF(s(x0)) -> c3(INF(s(s(x0)))) by 
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   EQ_2, TAKE_2, LENGTH_1, INF_1

Compound Symbols:   c1_1, c5_1, c7_1, c3_1


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

(41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) by 
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
Defined Rule Symbols:none

Defined Pair Symbols:   TAKE_2, LENGTH_1, INF_1, EQ_2

Compound Symbols:   c5_1, c7_1, c3_1, c1_1


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

(43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) by 
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, INF_1, EQ_2, TAKE_2

Compound Symbols:   c7_1, c3_1, c1_1, c5_1


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

(45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) by 
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
Defined Rule Symbols:none

Defined Pair Symbols:   INF_1, EQ_2, TAKE_2, LENGTH_1

Compound Symbols:   c3_1, c1_1, c5_1, c7_1


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

(47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1))) by 
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
Defined Rule Symbols:none

Defined Pair Symbols:   INF_1, TAKE_2, LENGTH_1, EQ_2

Compound Symbols:   c3_1, c5_1, c7_1, c1_1


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

(49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2))) by 
   TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3))))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
   TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3))))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
   TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3))))
Defined Rule Symbols:none

Defined Pair Symbols:   INF_1, LENGTH_1, EQ_2, TAKE_2

Compound Symbols:   c3_1, c7_1, c1_1, c5_1


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

(51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1))) by 
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c7(LENGTH(cons(z1, cons(y1, y2))))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
   TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c7(LENGTH(cons(z1, cons(y1, y2))))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
   TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c7(LENGTH(cons(z1, cons(y1, y2))))
Defined Rule Symbols:none

Defined Pair Symbols:   INF_1, EQ_2, TAKE_2, LENGTH_1

Compound Symbols:   c3_1, c1_1, c5_1, c7_1