KILLED
proof of input_DsbiDRrz3m.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 765 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 177 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtInstantiationProof [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
    (53) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) 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:

   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
   from(X) -> cons(X, from(s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
   prefix(L) -> cons(nil, zWadr(L, prefix(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:

   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
   from(X) -> cons(X, from(s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
   prefix(L) -> cons(nil, zWadr(L, prefix(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:

   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
   from(X) -> cons(X, from(s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
   prefix(L) -> cons(nil, zWadr(L, prefix(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:

   app(nil, YS) -> YS [1]
   app(cons(X, XS), YS) -> cons(X, app(XS, YS)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   zWadr(nil, YS) -> nil [1]
   zWadr(XS, nil) -> nil [1]
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS)) [1]
   prefix(L) -> cons(nil, zWadr(L, prefix(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:

   app(nil, YS) -> YS [1]
   app(cons(X, XS), YS) -> cons(X, app(XS, YS)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   zWadr(nil, YS) -> nil [1]
   zWadr(XS, nil) -> nil [1]
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS)) [1]
   prefix(L) -> cons(nil, zWadr(L, prefix(L))) [1]

The TRS has the following type information:
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
s :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
prefix :: nil:cons:s -> nil:cons: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:

   from_1

(c) The following functions are completely defined:

   prefix_1
   zWadr_2
   app_2

Due to the following rules being added:

   zWadr(v0, v1) -> nil [0]
   app(v0, v1) -> nil [0]

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:

   app(nil, YS) -> YS [1]
   app(cons(X, XS), YS) -> cons(X, app(XS, YS)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   zWadr(nil, YS) -> nil [1]
   zWadr(XS, nil) -> nil [1]
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS)) [1]
   prefix(L) -> cons(nil, zWadr(L, prefix(L))) [1]
   zWadr(v0, v1) -> nil [0]
   app(v0, v1) -> nil [0]

The TRS has the following type information:
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
s :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
prefix :: nil:cons:s -> nil:cons: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:

   app(nil, YS) -> YS [1]
   app(cons(X, XS), YS) -> cons(X, app(XS, YS)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   zWadr(nil, YS) -> nil [1]
   zWadr(XS, nil) -> nil [1]
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS)) [1]
   prefix(L) -> cons(nil, zWadr(L, cons(nil, zWadr(L, prefix(L))))) [2]
   zWadr(v0, v1) -> nil [0]
   app(v0, v1) -> nil [0]

The TRS has the following type information:
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
s :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
prefix :: nil:cons:s -> nil:cons: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:

   nil => 0

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

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

   app(z, z') -{ 1 }-> YS :|: z' = YS, YS >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 1 }-> 1 + X + app(XS, YS) :|: z' = YS, z = 1 + X + XS, X >= 0, YS >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(L, 1 + 0 + zWadr(L, prefix(L))) :|: z = L, L >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' = YS, YS >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z = XS, XS >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0


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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 1 }-> 1 + X + app(XS, z') :|: z = 1 + X + XS, X >= 0, z' >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0


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

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

   { from }
   { app }
   { zWadr }
   { prefix }

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 1 }-> 1 + X + app(XS, z') :|: z = 1 + X + XS, X >= 0, z' >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0

Function symbols to be analyzed: {from}, {app}, {zWadr}, {prefix}

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

(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:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 1 }-> 1 + X + app(XS, z') :|: z = 1 + X + XS, X >= 0, z' >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0

Function symbols to be analyzed: {from}, {app}, {zWadr}, {prefix}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: from
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:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 1 }-> 1 + X + app(XS, z') :|: z = 1 + X + XS, X >= 0, z' >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0

Function symbols to be analyzed: {from}, {app}, {zWadr}, {prefix}
Previous analysis results are:
  from: runtime: ?, size: O(1) [0]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 1 }-> 1 + X + app(XS, z') :|: z = 1 + X + XS, X >= 0, z' >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0

Function symbols to be analyzed: {from}, {app}, {zWadr}, {prefix}
Previous analysis results are:
  from: 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:

   app(v0, v1) -> null_app [0]
   zWadr(v0, v1) -> null_zWadr [0]

And the following fresh constants:    null_app, null_zWadr

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

(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:

   app(nil, YS) -> YS [1]
   app(cons(X, XS), YS) -> cons(X, app(XS, YS)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   zWadr(nil, YS) -> nil [1]
   zWadr(XS, nil) -> nil [1]
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS)) [1]
   prefix(L) -> cons(nil, zWadr(L, prefix(L))) [1]
   app(v0, v1) -> null_app [0]
   zWadr(v0, v1) -> null_zWadr [0]

The TRS has the following type information:
app :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
nil :: nil:cons:s:null_app:null_zWadr
cons :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
from :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
s :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
zWadr :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
prefix :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
null_app :: nil:cons:s:null_app:null_zWadr
null_zWadr :: nil:cons:s:null_app:null_zWadr

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:

   nil => 0
   null_app => 0
   null_zWadr => 0

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

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

   app(z, z') -{ 1 }-> YS :|: z' = YS, YS >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 1 }-> 1 + X + app(XS, YS) :|: z' = YS, z = 1 + X + XS, X >= 0, YS >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X
   prefix(z) -{ 1 }-> 1 + 0 + zWadr(L, prefix(L)) :|: z = L, L >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' = YS, YS >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z = XS, XS >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0

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:
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
Tuples:
   APP(nil, z0) -> c
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(nil, z0) -> c3
   ZWADR(z0, nil) -> c4
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
S tuples:
   APP(nil, z0) -> c
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(nil, z0) -> c3
   ZWADR(z0, nil) -> c4
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   app_2, from_1, zWadr_2, prefix_1

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c, c1_1, c2_1, c3, c4, c5_1, c6_1, c7_2


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

(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   ZWADR(z0, nil) -> c4
   APP(nil, z0) -> c
   ZWADR(nil, z0) -> c3

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   app_2, from_1, zWadr_2, prefix_1

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c1_1, c2_1, c5_1, c6_1, c7_2


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

(33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   from(z0) -> cons(z0, from(s(z0)))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c1_1, c2_1, c5_1, c6_1, c7_2


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

(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0)) by 
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, prefix(z0)))), PREFIX(z0))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, prefix(z0)))), PREFIX(z0))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, prefix(z0)))), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c1_1, c2_1, c5_1, c6_1, c7_2


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

(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, prefix(z0)))), PREFIX(z0)) by 
   PREFIX(nil) -> c7(ZWADR(nil, cons(nil, nil)), PREFIX(nil))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(nil) -> c7(ZWADR(nil, cons(nil, nil)), PREFIX(nil))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(nil) -> c7(ZWADR(nil, cons(nil, nil)), PREFIX(nil))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c1_1, c2_1, c5_1, c6_1, c7_2


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

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c1_1, c2_1, c5_1, c6_1, c7_2, c7_1


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

(41) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(z0) -> c2(FROM(s(z0))) by 
   FROM(s(x0)) -> c2(FROM(s(s(x0))))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(x0)) -> c2(FROM(s(s(x0))))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(x0)) -> c2(FROM(s(s(x0))))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, ZWADR_2, PREFIX_1, FROM_1

Compound Symbols:   c1_1, c5_1, c6_1, c7_2, c7_1, c2_1


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

(43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APP(cons(z0, z1), z2) -> c1(APP(z1, z2)) by 
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(x0)) -> c2(FROM(s(s(x0))))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
S tuples:
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(x0)) -> c2(FROM(s(s(x0))))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   ZWADR_2, PREFIX_1, FROM_1, APP_2

Compound Symbols:   c5_1, c6_1, c7_2, c7_1, c2_1, c1_1


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

(45) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(s(x0)) -> c2(FROM(s(s(x0)))) by 
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
S tuples:
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   ZWADR_2, PREFIX_1, APP_2, FROM_1

Compound Symbols:   c5_1, c6_1, c7_2, c7_1, c1_1, c2_1


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

(47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil))) by 
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
S tuples:
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   ZWADR_2, PREFIX_1, APP_2, FROM_1

Compound Symbols:   c6_1, c7_2, c7_1, c1_1, c2_1, c5_1


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

(49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3)) by 
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
S tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   PREFIX_1, APP_2, FROM_1, ZWADR_2

Compound Symbols:   c7_2, c7_1, c1_1, c2_1, c5_1, c6_1


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

(51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2)) by 
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
S tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   PREFIX_1, FROM_1, ZWADR_2, APP_2

Compound Symbols:   c7_2, c7_1, c2_1, c5_1, c6_1, c1_1


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

(53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil))) by 
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
S tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   PREFIX_1, FROM_1, ZWADR_2, APP_2

Compound Symbols:   c7_2, c7_1, c2_1, c6_1, c1_1, c5_1


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

(55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3))) by 
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(y4, y5)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(y4, y5))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(y2, cons(y3, cons(y4, y5))), z5))) -> c6(ZWADR(cons(z1, z2), cons(cons(y2, cons(y3, cons(y4, y5))), z5)))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(y4, y5)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(y4, y5))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(y2, cons(y3, cons(y4, y5))), z5))) -> c6(ZWADR(cons(z1, z2), cons(cons(y2, cons(y3, cons(y4, y5))), z5)))
S tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(y4, y5)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(y4, y5))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(y2, cons(y3, cons(y4, y5))), z5))) -> c6(ZWADR(cons(z1, z2), cons(cons(y2, cons(y3, cons(y4, y5))), z5)))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   PREFIX_1, FROM_1, ZWADR_2, APP_2

Compound Symbols:   c7_2, c7_1, c2_1, c6_1, c1_1, c5_1


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

(57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5))) by 
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(z4, cons(z5, cons(y4, y5))), z7))) -> c6(ZWADR(cons(z1, z2), cons(cons(z4, cons(z5, cons(y4, y5))), z7)))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7)))))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9)))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8))))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(y4, y5)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(y4, y5))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(y2, cons(y3, cons(y4, y5))), z5))) -> c6(ZWADR(cons(z1, z2), cons(cons(y2, cons(y3, cons(y4, y5))), z5)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7)))))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9)))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8))))
S tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(y4, y5)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(y4, y5))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(y2, cons(y3, cons(y4, y5))), z5))) -> c6(ZWADR(cons(z1, z2), cons(cons(y2, cons(y3, cons(y4, y5))), z5)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7)))))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9)))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8))))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   PREFIX_1, FROM_1, APP_2, ZWADR_2

Compound Symbols:   c7_2, c7_1, c2_1, c1_1, c5_1, c6_1