MAYBE
proof of input_CJRmiko6Jy.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) InliningProof [UPPER BOUND(ID), 121 ms]
        (20) CpxRNTS
        (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 62 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 851 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 253 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 39 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 62 ms]
        (42) CpxRNTS
        (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 1045 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
        (48) CpxRNTS
(49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(50) CdtProblem
    (51) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (52) CdtProblem
    (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 7 ms]
    (60) CdtProblem
    (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 45 ms]
    (62) CdtProblem
    (63) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) 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:

   zeros -> cons(0, n__zeros)
   and(tt, X) -> activate(X)
   length(nil) -> 0
   length(cons(N, L)) -> s(length(activate(L)))
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL)))
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(X) -> 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:

   zeros -> cons(0', n__zeros)
   and(tt, X) -> activate(X)
   length(nil) -> 0'
   length(cons(N, L)) -> s(length(activate(L)))
   take(0', IL) -> nil
   take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL)))
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(X) -> 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:

   zeros -> cons(0, n__zeros)
   and(tt, X) -> activate(X)
   length(nil) -> 0
   length(cons(N, L)) -> s(length(activate(L)))
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL)))
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(X) -> 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:

   zeros -> cons(0, n__zeros) [1]
   and(tt, X) -> activate(X) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> s(length(activate(L))) [1]
   take(0, IL) -> nil [1]
   take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) [1]
   zeros -> n__zeros [1]
   take(X1, X2) -> n__take(X1, X2) [1]
   activate(n__zeros) -> zeros [1]
   activate(n__take(X1, X2)) -> take(X1, X2) [1]
   activate(X) -> 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:

   zeros -> cons(0, n__zeros) [1]
   and(tt, X) -> activate(X) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> s(length(activate(L))) [1]
   take(0, IL) -> nil [1]
   take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) [1]
   zeros -> n__zeros [1]
   take(X1, X2) -> n__take(X1, X2) [1]
   activate(n__zeros) -> zeros [1]
   activate(n__take(X1, X2)) -> take(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
zeros :: n__zeros:cons:nil:n__take
cons :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
0 :: 0:s
n__zeros :: n__zeros:cons:nil:n__take
and :: tt -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
tt :: tt
activate :: n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
length :: n__zeros:cons:nil:n__take -> 0:s
nil :: n__zeros:cons:nil:n__take
s :: 0:s -> 0:s
take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
n__take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take

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:

   length(v0) -> null_length [0]

And the following fresh constants:    null_length

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

(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:

   zeros -> cons(0, n__zeros) [1]
   and(tt, X) -> activate(X) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> s(length(activate(L))) [1]
   take(0, IL) -> nil [1]
   take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) [1]
   zeros -> n__zeros [1]
   take(X1, X2) -> n__take(X1, X2) [1]
   activate(n__zeros) -> zeros [1]
   activate(n__take(X1, X2)) -> take(X1, X2) [1]
   activate(X) -> X [1]
   length(v0) -> null_length [0]

The TRS has the following type information:
zeros :: n__zeros:cons:nil:n__take
cons :: 0:s:null_length -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
0 :: 0:s:null_length
n__zeros :: n__zeros:cons:nil:n__take
and :: tt -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
tt :: tt
activate :: n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
length :: n__zeros:cons:nil:n__take -> 0:s:null_length
nil :: n__zeros:cons:nil:n__take
s :: 0:s:null_length -> 0:s:null_length
take :: 0:s:null_length -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
n__take :: 0:s:null_length -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
null_length :: 0:s:null_length

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:

   0 => 0
   n__zeros => 0
   tt => 0
   nil => 1
   null_length => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> zeros :|: z = 0
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   length(z) -{ 1 }-> 1 + length(activate(L)) :|: z = 1 + N + L, L >= 0, N >= 0
   take(z, z') -{ 1 }-> 1 :|: z' = IL, z = 0, IL >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + M + activate(IL)) :|: z = 1 + M, M >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

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:

   and_2
   length_1

(c) The following functions are completely defined:

   activate_1
   take_2
   zeros

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   zeros -> cons(0, n__zeros) [1]
   and(tt, X) -> activate(X) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> s(length(activate(L))) [1]
   take(0, IL) -> nil [1]
   take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) [1]
   zeros -> n__zeros [1]
   take(X1, X2) -> n__take(X1, X2) [1]
   activate(n__zeros) -> zeros [1]
   activate(n__take(X1, X2)) -> take(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
zeros :: n__zeros:cons:nil:n__take
cons :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
0 :: 0:s
n__zeros :: n__zeros:cons:nil:n__take
and :: tt -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
tt :: tt
activate :: n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
length :: n__zeros:cons:nil:n__take -> 0:s
nil :: n__zeros:cons:nil:n__take
s :: 0:s -> 0:s
take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
n__take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take

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:

   zeros -> cons(0, n__zeros) [1]
   and(tt, X) -> activate(X) [1]
   length(nil) -> 0 [1]
   length(cons(N, n__zeros)) -> s(length(zeros)) [2]
   length(cons(N, n__take(X1', X2'))) -> s(length(take(X1', X2'))) [2]
   length(cons(N, L)) -> s(length(L)) [2]
   take(0, IL) -> nil [1]
   take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) [1]
   zeros -> n__zeros [1]
   take(X1, X2) -> n__take(X1, X2) [1]
   activate(n__zeros) -> zeros [1]
   activate(n__take(X1, X2)) -> take(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
zeros :: n__zeros:cons:nil:n__take
cons :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
0 :: 0:s
n__zeros :: n__zeros:cons:nil:n__take
and :: tt -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
tt :: tt
activate :: n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
length :: n__zeros:cons:nil:n__take -> 0:s
nil :: n__zeros:cons:nil:n__take
s :: 0:s -> 0:s
take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take
n__take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take

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:

   0 => 0
   n__zeros => 0
   tt => 0
   nil => 1

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> zeros :|: z = 0
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(zeros) :|: z = 1 + N + 0, N >= 0
   length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   take(z, z') -{ 1 }-> 1 :|: z' = IL, z = 0, IL >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + M + activate(IL)) :|: z = 1 + M, M >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 


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

(19) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   zeros -{ 1 }-> 1 + 0 + 0 :|: 
   zeros -{ 1 }-> 0 :|: 

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z = 1 + N + 0, N >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 1 + N + 0, N >= 0
   take(z, z') -{ 1 }-> 1 :|: z' = IL, z = 0, IL >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + M + activate(IL)) :|: z = 1 + M, M >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 


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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 


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

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

   { zeros }
   { take, activate }
   { and }
   { length }

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {zeros}, {take,activate}, {and}, {length}

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

(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:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {zeros}, {take,activate}, {and}, {length}

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: zeros
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:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: zeros
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:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {take,activate}, {and}, {length}
Previous analysis results are:
  zeros: 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:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {take,activate}, {and}, {length}
Previous analysis results are:
  zeros: runtime: O(1) [1], size: O(1) [1]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {take,activate}, {and}, {length}
Previous analysis results are:
  zeros: runtime: O(1) [1], size: O(1) [1]
  take: runtime: ?, size: O(n^1) [1 + z + z']
  activate: runtime: ?, size: O(n^1) [1 + z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: take
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 7 + 2*z + 2*z'

Computed RUNTIME bound using CoFloCo for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 6 + 2*z

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {and}, {length}
Previous analysis results are:
  zeros: runtime: O(1) [1], size: O(1) [1]
  take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z']
  activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z]

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

(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:

   activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {and}, {length}
Previous analysis results are:
  zeros: runtime: O(1) [1], size: O(1) [1]
  take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z']
  activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {and}, {length}
Previous analysis results are:
  zeros: runtime: O(1) [1], size: O(1) [1]
  take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z']
  activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z]
  and: runtime: ?, size: O(n^1) [1 + z']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: and
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 7 + 2*z'

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

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

   activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {length}
Previous analysis results are:
  zeros: runtime: O(1) [1], size: O(1) [1]
  take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z']
  activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z]
  and: runtime: O(n^1) [7 + 2*z'], size: O(n^1) [1 + z']

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

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {length}
Previous analysis results are:
  zeros: runtime: O(1) [1], size: O(1) [1]
  take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z']
  activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z]
  and: runtime: O(n^1) [7 + 2*z'], size: O(n^1) [1 + z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {length}
Previous analysis results are:
  zeros: runtime: O(1) [1], size: O(1) [1]
  take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z']
  activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z]
  and: runtime: O(n^1) [7 + 2*z'], size: O(n^1) [1 + z']
  length: runtime: ?, size: INF

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0
   length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0
   take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0
   take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0
   take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {length}
Previous analysis results are:
  zeros: runtime: O(1) [1], size: O(1) [1]
  take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z']
  activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z]
  and: runtime: O(n^1) [7 + 2*z'], size: O(n^1) [1 + z']
  length: runtime: INF, size: INF

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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   and(tt, z0) -> activate(z0)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(activate(z1)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
Tuples:
   ZEROS -> c
   ZEROS -> c1
   AND(tt, z0) -> c2(ACTIVATE(z0))
   LENGTH(nil) -> c3
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
   TAKE(0, z0) -> c5
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   TAKE(z0, z1) -> c7
   ACTIVATE(n__zeros) -> c8(ZEROS)
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   ACTIVATE(z0) -> c10
S tuples:
   ZEROS -> c
   ZEROS -> c1
   AND(tt, z0) -> c2(ACTIVATE(z0))
   LENGTH(nil) -> c3
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
   TAKE(0, z0) -> c5
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   TAKE(z0, z1) -> c7
   ACTIVATE(n__zeros) -> c8(ZEROS)
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   ACTIVATE(z0) -> c10
K tuples:none
Defined Rule Symbols:   zeros, and_2, length_1, take_2, activate_1

Defined Pair Symbols:   ZEROS, AND_2, LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c, c1, c2_1, c3, c4_2, c5, c6_1, c7, c8_1, c9_1, c10


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

(51) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   AND(tt, z0) -> c2(ACTIVATE(z0))
Removed 7 trailing nodes:
   TAKE(0, z0) -> c5
   LENGTH(nil) -> c3
   ZEROS -> c1
   ACTIVATE(n__zeros) -> c8(ZEROS)
   ZEROS -> c
   ACTIVATE(z0) -> c10
   TAKE(z0, z1) -> c7

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   and(tt, z0) -> activate(z0)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(activate(z1)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
Tuples:
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
S tuples:
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
K tuples:none
Defined Rule Symbols:   zeros, and_2, length_1, take_2, activate_1

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c6_1, c9_1


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

(53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   and(tt, z0) -> activate(z0)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(activate(z1)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
S tuples:
   LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
K tuples:none
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c6_1, c9_1


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1)) by 
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros), ACTIVATE(n__zeros))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros), ACTIVATE(n__zeros))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
S tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros), ACTIVATE(n__zeros))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   TAKE_2, ACTIVATE_1, LENGTH_1

Compound Symbols:   c6_1, c9_1, c4_2


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

(57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
S tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
K tuples:none
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   TAKE_2, ACTIVATE_1, LENGTH_1

Compound Symbols:   c6_1, c9_1, c4_2, c4_1


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

(59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
We considered the (Usable) Rules:
   take(z0, z1) -> n__take(z0, z1)
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(0, z0) -> nil
   zeros -> n__zeros
   zeros -> cons(0, n__zeros)
And the Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [3]
   POL(ACTIVATE(x_1)) = 0
   POL(LENGTH(x_1)) = x_1
   POL(TAKE(x_1, x_2)) = 0
   POL(activate(x_1)) = [3]
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(n__take(x_1, x_2)) = x_1
   POL(n__zeros) = 0
   POL(nil) = [3]
   POL(s(x_1)) = [1] + x_1
   POL(take(x_1, x_2)) = x_1
   POL(zeros) = [1]

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
S tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
K tuples:
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   TAKE_2, ACTIVATE_1, LENGTH_1

Compound Symbols:   c6_1, c9_1, c4_2, c4_1


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

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
We considered the (Usable) Rules:
   take(z0, z1) -> n__take(z0, z1)
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   activate(n__take(z0, z1)) -> take(z0, z1)
   take(0, z0) -> nil
   zeros -> n__zeros
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   activate(n__zeros) -> zeros
And the Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ACTIVATE(x_1)) = [2]x_1
   POL(LENGTH(x_1)) = x_1^2
   POL(TAKE(x_1, x_2)) = x_1 + [2]x_2
   POL(activate(x_1)) = [2] + x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(n__take(x_1, x_2)) = x_1 + x_2
   POL(n__zeros) = [1]
   POL(nil) = [1]
   POL(s(x_1)) = [2] + x_1
   POL(take(x_1, x_2)) = x_1 + x_2
   POL(zeros) = [2]

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
S tuples:
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
K tuples:
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   TAKE_2, ACTIVATE_1, LENGTH_1

Compound Symbols:   c6_1, c9_1, c4_2, c4_1


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

(63) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
K tuples:
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   TAKE_2, ACTIVATE_1, LENGTH_1

Compound Symbols:   c6_1, c9_1, c4_2, c4_1


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) by 
   LENGTH(cons(x0, n__take(0, z0))) -> c4(LENGTH(nil), ACTIVATE(n__take(0, z0)))
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(n__take(z0, z1)), ACTIVATE(n__take(z0, z1)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
   LENGTH(cons(x0, n__take(0, z0))) -> c4(LENGTH(nil), ACTIVATE(n__take(0, z0)))
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(n__take(z0, z1)), ACTIVATE(n__take(z0, z1)))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
K tuples:
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   TAKE_2, ACTIVATE_1, LENGTH_1

Compound Symbols:   c6_1, c9_1, c4_2, c4_1


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

(67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0)))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1)))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros))
K tuples:
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   TAKE_2, ACTIVATE_1, LENGTH_1

Compound Symbols:   c6_1, c9_1, c4_2, c4_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) by 
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0)))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros))
K tuples:
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   TAKE_2, ACTIVATE_1, LENGTH_1

Compound Symbols:   c6_1, c9_1, c4_2, c4_1


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

(71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0)))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   TAKE_2, ACTIVATE_1, LENGTH_1

Compound Symbols:   c6_1, c9_1, c4_2, c4_1


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

(73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) by 
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0)))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1))
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   ACTIVATE_1, LENGTH_1, TAKE_2

Compound Symbols:   c9_1, c4_2, c4_1, c6_1


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

(75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) by 
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0)))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c4_1, c6_1, c9_1


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

(77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0))
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c4_1, c6_1, c9_1


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

(79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) by 
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)), ACTIVATE(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))), ACTIVATE(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))), ACTIVATE(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)), ACTIVATE(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(LENGTH(n__take(s(y0), cons(y1, n__take(y2, y3)))), ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)), ACTIVATE(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))), ACTIVATE(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))), ACTIVATE(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)), ACTIVATE(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(LENGTH(n__take(s(y0), cons(y1, n__take(y2, y3)))), ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)), ACTIVATE(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))), ACTIVATE(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))), ACTIVATE(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)), ACTIVATE(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(LENGTH(n__take(s(y0), cons(y1, n__take(y2, y3)))), ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c4_1, c6_1, c9_1


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

(81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1)))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c4_1, c6_1, c9_1


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

(83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1))) by 
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c4_1, c6_1, c9_1


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

(85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) by 
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3))))
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, ACTIVATE_1, TAKE_2

Compound Symbols:   c4_2, c4_1, c9_1, c6_1


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

(87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) by 
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c4_1, c6_1, c9_1


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

(89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) by 
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(z1, n__zeros))) -> c4(LENGTH(cons(z1, n__zeros)))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4))))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3)))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4))))))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4))))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3)))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4))))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4))))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3)))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4))))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c4_1, c6_1, c9_1


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

(91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) by 
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4))))))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4))))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3)))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4))))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4))))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3)))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4))))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c4_1, c6_1, c9_1


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

(93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) by 
   LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5)))))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4))))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3)))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4))))))
   LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5)))))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4))))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3)))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4))))))
   LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5)))))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c4_1, c6_1, c9_1


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(95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) by 
   LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5)))))))

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(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2))))
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4))))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3)))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4))))))
   LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5)))))))
S tuples:
   LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros)))
K tuples:
   LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)))
   TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))))
   ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5))))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4))))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3)))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4))))))
   LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5)))))))
Defined Rule Symbols:   activate_1, zeros, take_2

Defined Pair Symbols:   LENGTH_1, TAKE_2, ACTIVATE_1

Compound Symbols:   c4_2, c4_1, c6_1, c9_1