MAYBE
proof of input_THdKtk6nEZ.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 805 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
        (24) CpxRNTS
    (25) CompletionProof [UPPER BOUND(ID), 0 ms]
    (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
(29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(30) CdtProblem
    (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 74 ms]
    (36) CdtProblem
    (37) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (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), 1 ms]
    (58) CdtProblem
    (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) 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:

   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   dbls(nil) -> nil
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)
   indx(nil, X) -> nil
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
   from(X) -> cons(X, from(s(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:

   dbl(0') -> 0'
   dbl(s(X)) -> s(s(dbl(X)))
   dbls(nil) -> nil
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
   sel(0', cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)
   indx(nil, X) -> nil
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
   from(X) -> cons(X, from(s(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:

   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   dbls(nil) -> nil
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)
   indx(nil, X) -> nil
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
   from(X) -> cons(X, from(s(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:

   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   dbls(nil) -> nil [1]
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [1]
   indx(nil, X) -> nil [1]
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) [1]
   from(X) -> cons(X, from(s(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:

   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   dbls(nil) -> nil [1]
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [1]
   indx(nil, X) -> nil [1]
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) [1]
   from(X) -> cons(X, from(s(X))) [1]

The TRS has the following type information:
dbl :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
dbls :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
sel :: 0:s -> nil:cons -> 0:s
indx :: nil:cons -> nil:cons -> nil:cons
from :: 0:s -> nil:cons

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:

   dbl_1
   dbls_1
   sel_2
   indx_2
   from_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   dbls(nil) -> nil [1]
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [1]
   indx(nil, X) -> nil [1]
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) [1]
   from(X) -> cons(X, from(s(X))) [1]

The TRS has the following type information:
dbl :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
dbls :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
sel :: 0:s -> nil:cons -> 0:s
indx :: nil:cons -> nil:cons -> nil:cons
from :: 0:s -> nil:cons

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:

   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   dbls(nil) -> nil [1]
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [1]
   indx(nil, X) -> nil [1]
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) [1]
   from(X) -> cons(X, from(s(X))) [1]

The TRS has the following type information:
dbl :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
dbls :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
sel :: 0:s -> nil:cons -> 0:s
indx :: nil:cons -> nil:cons -> nil:cons
from :: 0:s -> nil:cons

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

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

   0 => 0
   nil => 0

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

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

   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0
   dbls(z) -{ 1 }-> 0 :|: z = 0
   dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X
   indx(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   indx(z, z') -{ 1 }-> 1 + sel(X, Z) + indx(Y, Z) :|: Z >= 0, Y >= 0, z = 1 + X + Y, X >= 0, z' = Z
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z


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

(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:

   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   dbls(z) -{ 1 }-> 0 :|: z = 0
   dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   indx(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   indx(z, z') -{ 1 }-> 1 + sel(X, z') + indx(Y, z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z


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

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

   { from }
   { sel }
   { dbl }
   { indx }
   { dbls }

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

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

   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   dbls(z) -{ 1 }-> 0 :|: z = 0
   dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   indx(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   indx(z, z') -{ 1 }-> 1 + sel(X, z') + indx(Y, z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z

Function symbols to be analyzed: {from}, {sel}, {dbl}, {indx}, {dbls}

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

(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:

   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   dbls(z) -{ 1 }-> 0 :|: z = 0
   dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   indx(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   indx(z, z') -{ 1 }-> 1 + sel(X, z') + indx(Y, z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z

Function symbols to be analyzed: {from}, {sel}, {dbl}, {indx}, {dbls}

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

(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:

   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   dbls(z) -{ 1 }-> 0 :|: z = 0
   dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   indx(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   indx(z, z') -{ 1 }-> 1 + sel(X, z') + indx(Y, z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z

Function symbols to be analyzed: {from}, {sel}, {dbl}, {indx}, {dbls}
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:

   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   dbls(z) -{ 1 }-> 0 :|: z = 0
   dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   indx(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   indx(z, z') -{ 1 }-> 1 + sel(X, z') + indx(Y, z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z

Function symbols to be analyzed: {from}, {sel}, {dbl}, {indx}, {dbls}
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:

   sel(v0, v1) -> null_sel [0]
   dbl(v0) -> null_dbl [0]

And the following fresh constants:    null_sel, null_dbl

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

(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:

   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   dbls(nil) -> nil [1]
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [1]
   indx(nil, X) -> nil [1]
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   sel(v0, v1) -> null_sel [0]
   dbl(v0) -> null_dbl [0]

The TRS has the following type information:
dbl :: 0:s:null_sel:null_dbl -> 0:s:null_sel:null_dbl
0 :: 0:s:null_sel:null_dbl
s :: 0:s:null_sel:null_dbl -> 0:s:null_sel:null_dbl
dbls :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s:null_sel:null_dbl -> nil:cons -> nil:cons
sel :: 0:s:null_sel:null_dbl -> nil:cons -> 0:s:null_sel:null_dbl
indx :: nil:cons -> nil:cons -> nil:cons
from :: 0:s:null_sel:null_dbl -> nil:cons
null_sel :: 0:s:null_sel:null_dbl
null_dbl :: 0:s:null_sel:null_dbl

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

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

   0 => 0
   nil => 0
   null_sel => 0
   null_dbl => 0

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

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

   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0
   dbls(z) -{ 1 }-> 0 :|: z = 0
   dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X
   indx(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   indx(z, z') -{ 1 }-> 1 + sel(X, Z) + indx(Y, Z) :|: Z >= 0, Y >= 0, z = 1 + X + Y, X >= 0, z' = Z
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

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:
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
   dbls(nil) -> nil
   dbls(cons(z0, z1)) -> cons(dbl(z0), dbls(z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)
   indx(nil, z0) -> nil
   indx(cons(z0, z1), z2) -> cons(sel(z0, z2), indx(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   DBL(0) -> c
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(nil) -> c2
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(0, cons(z0, z1)) -> c5
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(nil, z0) -> c7
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(z0) -> c10(FROM(s(z0)))
S tuples:
   DBL(0) -> c
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(nil) -> c2
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(0, cons(z0, z1)) -> c5
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(nil, z0) -> c7
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(z0) -> c10(FROM(s(z0)))
K tuples:none
Defined Rule Symbols:   dbl_1, dbls_1, sel_2, indx_2, from_1

Defined Pair Symbols:   DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1

Compound Symbols:   c, c1_1, c2, c3_1, c4_1, c5, c6_1, c7, c8_1, c9_1, c10_1


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

(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   INDX(nil, z0) -> c7
   DBLS(nil) -> c2
   DBL(0) -> c
   SEL(0, cons(z0, z1)) -> c5

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
   dbls(nil) -> nil
   dbls(cons(z0, z1)) -> cons(dbl(z0), dbls(z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)
   indx(nil, z0) -> nil
   indx(cons(z0, z1), z2) -> cons(sel(z0, z2), indx(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(z0) -> c10(FROM(s(z0)))
S tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(z0) -> c10(FROM(s(z0)))
K tuples:none
Defined Rule Symbols:   dbl_1, dbls_1, sel_2, indx_2, from_1

Defined Pair Symbols:   DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1

Compound Symbols:   c1_1, c3_1, c4_1, c6_1, c8_1, c9_1, c10_1


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

(33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
   dbls(nil) -> nil
   dbls(cons(z0, z1)) -> cons(dbl(z0), dbls(z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)
   indx(nil, z0) -> nil
   indx(cons(z0, z1), z2) -> cons(sel(z0, z2), indx(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(z0) -> c10(FROM(s(z0)))
S tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(z0) -> c10(FROM(s(z0)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1

Compound Symbols:   c1_1, c3_1, c4_1, c6_1, c8_1, c9_1, c10_1


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

(35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(z0) -> c10(FROM(s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(DBL(x_1)) = [1] + x_1
   POL(DBLS(x_1)) = [1] + x_1
   POL(FROM(x_1)) = 0
   POL(INDX(x_1, x_2)) = x_1 + x_2
   POL(SEL(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1] + x_1

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(z0) -> c10(FROM(s(z0)))
S tuples:
   FROM(z0) -> c10(FROM(s(z0)))
K tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1

Compound Symbols:   c1_1, c3_1, c4_1, c6_1, c8_1, c9_1, c10_1


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

(37) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(z0) -> c10(FROM(s(z0))) by 
   FROM(s(x0)) -> c10(FROM(s(s(x0))))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(s(x0)) -> c10(FROM(s(s(x0))))
S tuples:
   FROM(s(x0)) -> c10(FROM(s(s(x0))))
K tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1

Compound Symbols:   c1_1, c3_1, c4_1, c6_1, c8_1, c9_1, c10_1


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

(39) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(s(x0)) -> c10(FROM(s(s(x0)))) by 
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   DBL(s(z0)) -> c1(DBL(z0))
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1

Compound Symbols:   c1_1, c3_1, c4_1, c6_1, c8_1, c9_1, c10_1


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

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   DBLS(cons(z0, z1)) -> c3(DBL(z0))
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
Defined Rule Symbols:none

Defined Pair Symbols:   DBLS_1, SEL_2, INDX_2, FROM_1, DBL_1

Compound Symbols:   c3_1, c4_1, c6_1, c8_1, c9_1, c10_1, c1_1


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

(43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DBLS(cons(z0, z1)) -> c3(DBL(z0)) by 
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   DBLS(cons(z0, z1)) -> c4(DBLS(z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
Defined Rule Symbols:none

Defined Pair Symbols:   DBLS_1, SEL_2, INDX_2, FROM_1, DBL_1

Compound Symbols:   c4_1, c6_1, c8_1, c9_1, c10_1, c1_1, c3_1


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

(45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DBLS(cons(z0, z1)) -> c4(DBLS(z1)) by 
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
Defined Rule Symbols:none

Defined Pair Symbols:   SEL_2, INDX_2, FROM_1, DBL_1, DBLS_1

Compound Symbols:   c6_1, c8_1, c9_1, c10_1, c1_1, c3_1, c4_1


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

(47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) by 
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2))
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
Defined Rule Symbols:none

Defined Pair Symbols:   INDX_2, FROM_1, DBL_1, DBLS_1, SEL_2

Compound Symbols:   c8_1, c9_1, c10_1, c1_1, c3_1, c4_1, c6_1


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

(49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) by 
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
Defined Rule Symbols:none

Defined Pair Symbols:   INDX_2, FROM_1, DBL_1, DBLS_1, SEL_2

Compound Symbols:   c9_1, c10_1, c1_1, c3_1, c4_1, c6_1, c8_1


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

(51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) by 
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   DBL(s(s(y0))) -> c1(DBL(s(y0)))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1, DBL_1, DBLS_1, SEL_2, INDX_2

Compound Symbols:   c10_1, c1_1, c3_1, c4_1, c6_1, c8_1, c9_1


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

(53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DBL(s(s(y0))) -> c1(DBL(s(y0))) by 
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1, DBLS_1, SEL_2, INDX_2, DBL_1

Compound Symbols:   c10_1, c3_1, c4_1, c6_1, c8_1, c9_1, c1_1


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

(55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) by 
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1)))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1, DBLS_1, SEL_2, INDX_2, DBL_1

Compound Symbols:   c10_1, c4_1, c6_1, c8_1, c9_1, c1_1, c3_1


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

(57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) by 
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1, DBLS_1, SEL_2, INDX_2, DBL_1

Compound Symbols:   c10_1, c4_1, c6_1, c8_1, c9_1, c1_1, c3_1


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

(59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) by 
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1, SEL_2, INDX_2, DBL_1, DBLS_1

Compound Symbols:   c10_1, c6_1, c8_1, c9_1, c1_1, c3_1, c4_1


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

(61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) by 
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1, INDX_2, DBL_1, DBLS_1, SEL_2

Compound Symbols:   c10_1, c8_1, c9_1, c1_1, c3_1, c4_1, c6_1


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) by 
   INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4)))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
   INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4)))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
   INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4)))))
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1, INDX_2, DBL_1, DBLS_1, SEL_2

Compound Symbols:   c10_1, c9_1, c1_1, c3_1, c4_1, c6_1, c8_1


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

(65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) by 
   INDX(cons(z0, cons(z1, cons(y1, y2))), z3) -> c9(INDX(cons(z1, cons(y1, y2)), z3))
   INDX(cons(z0, cons(z1, cons(s(s(y1)), y2))), cons(y3, cons(y4, y5))) -> c9(INDX(cons(z1, cons(s(s(y1)), y2)), cons(y3, cons(y4, y5))))
   INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(y2, cons(y3, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(y2, cons(y3, cons(y4, y5)))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
   INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4)))))
   INDX(cons(z0, cons(z1, cons(y1, y2))), z3) -> c9(INDX(cons(z1, cons(y1, y2)), z3))
   INDX(cons(z0, cons(z1, cons(s(s(y1)), y2))), cons(y3, cons(y4, y5))) -> c9(INDX(cons(z1, cons(s(s(y1)), y2)), cons(y3, cons(y4, y5))))
   INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(y2, cons(y3, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(y2, cons(y3, cons(y4, y5)))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
   INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4)))))
   INDX(cons(z0, cons(z1, cons(y1, y2))), z3) -> c9(INDX(cons(z1, cons(y1, y2)), z3))
   INDX(cons(z0, cons(z1, cons(s(s(y1)), y2))), cons(y3, cons(y4, y5))) -> c9(INDX(cons(z1, cons(s(s(y1)), y2)), cons(y3, cons(y4, y5))))
   INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(y2, cons(y3, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(y2, cons(y3, cons(y4, y5)))))
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1, INDX_2, DBL_1, DBLS_1, SEL_2

Compound Symbols:   c10_1, c9_1, c1_1, c3_1, c4_1, c6_1, c8_1


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

(67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) by 
   INDX(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(s(s(y1)), y2)), cons(z3, cons(z4, z5))))
   INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(z3, cons(z4, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(z3, cons(z4, cons(y4, y5)))))
   INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(y2, y3))), cons(z3, cons(z4, z5))))
   INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))), cons(z3, cons(z4, z5))))
   INDX(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2))), cons(z3, cons(z4, cons(y5, y6)))) -> c9(INDX(cons(s(s(z1)), cons(s(s(s(y1))), y2)), cons(z3, cons(z4, cons(y5, y6)))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
   INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4)))))
   INDX(cons(z0, cons(z1, cons(y1, y2))), z3) -> c9(INDX(cons(z1, cons(y1, y2)), z3))
   INDX(cons(z0, cons(z1, cons(s(s(y1)), y2))), cons(y3, cons(y4, y5))) -> c9(INDX(cons(z1, cons(s(s(y1)), y2)), cons(y3, cons(y4, y5))))
   INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(y2, cons(y3, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(y2, cons(y3, cons(y4, y5)))))
   INDX(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(s(s(y1)), y2)), cons(z3, cons(z4, z5))))
   INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(y2, y3))), cons(z3, cons(z4, z5))))
   INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))), cons(z3, cons(z4, z5))))
   INDX(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2))), cons(z3, cons(z4, cons(y5, y6)))) -> c9(INDX(cons(s(s(z1)), cons(s(s(s(y1))), y2)), cons(z3, cons(z4, cons(y5, y6)))))
S tuples:
   FROM(s(s(x0))) -> c10(FROM(s(s(s(x0)))))
K tuples:
   DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0))))
   DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0)))))
   DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2))))
   DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2)))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))))
   DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
   INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4)))))
   INDX(cons(z0, cons(z1, cons(y1, y2))), z3) -> c9(INDX(cons(z1, cons(y1, y2)), z3))
   INDX(cons(z0, cons(z1, cons(s(s(y1)), y2))), cons(y3, cons(y4, y5))) -> c9(INDX(cons(z1, cons(s(s(y1)), y2)), cons(y3, cons(y4, y5))))
   INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(y2, cons(y3, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(y2, cons(y3, cons(y4, y5)))))
   INDX(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(s(s(y1)), y2)), cons(z3, cons(z4, z5))))
   INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(y2, y3))), cons(z3, cons(z4, z5))))
   INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))), cons(z3, cons(z4, z5))))
   INDX(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2))), cons(z3, cons(z4, cons(y5, y6)))) -> c9(INDX(cons(s(s(z1)), cons(s(s(s(y1))), y2)), cons(z3, cons(z4, cons(y5, y6)))))
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1, DBL_1, DBLS_1, SEL_2, INDX_2

Compound Symbols:   c10_1, c1_1, c3_1, c4_1, c6_1, c8_1, c9_1