KILLED
proof of input_T1EdOsr30l.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 315 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 50 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 3649 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 904 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (38) CdtProblem
    (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 109 ms]
    (42) CdtProblem
    (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 7 ms]
    (148) CdtProblem
    (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) 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:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs')))
   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil)
   binom(Nil, k) -> Cons(Nil, Nil)
   goal(x, y) -> binom(x, y)

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:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs')))
   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil)
   binom(Nil, k) -> Cons(Nil, Nil)
   goal(x, y) -> binom(x, y)

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:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs')))
   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil)
   binom(Nil, k) -> Cons(Nil, Nil)
   goal(x, y) -> binom(x, y)

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:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1]
   @(Nil, ys) -> ys [1]
   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) [1]
   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) [1]
   binom(Nil, k) -> Cons(Nil, Nil) [1]
   goal(x, y) -> binom(x, y) [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:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1]
   @(Nil, ys) -> ys [1]
   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) [1]
   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) [1]
   binom(Nil, k) -> Cons(Nil, Nil) [1]
   goal(x, y) -> binom(x, y) [1]

The TRS has the following type information:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
binom :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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:

   goal_2

(c) The following functions are completely defined:

   binom_2
   @_2

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:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1]
   @(Nil, ys) -> ys [1]
   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) [1]
   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) [1]
   binom(Nil, k) -> Cons(Nil, Nil) [1]
   goal(x, y) -> binom(x, y) [1]

The TRS has the following type information:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
binom :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1]
   @(Nil, ys) -> ys [1]
   binom(Cons(x, Cons(x'', xs'')), Cons(x', Cons(x''', xs'''))) -> @(@(binom(xs'', xs'''), binom(xs'', Cons(x''', xs'''))), @(binom(xs'', Cons(x''', xs''')), binom(xs'', Cons(x', Cons(x''', xs'''))))) [3]
   binom(Cons(x, Cons(x1, xs1)), Cons(x', Nil)) -> @(Cons(Nil, Nil), @(binom(xs1, Nil), binom(xs1, Cons(x', Nil)))) [3]
   binom(Cons(x, Nil), Cons(x', xs')) -> @(Cons(Nil, Nil), Cons(Nil, Nil)) [3]
   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) [1]
   binom(Nil, k) -> Cons(Nil, Nil) [1]
   goal(x, y) -> binom(x, y) [1]

The TRS has the following type information:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
binom :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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:

   @(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0
   @(z, z') -{ 1 }-> 1 + x + @(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0
   binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''')
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + x' + 0))) :|: x1 >= 0, z' = 1 + x' + 0, x >= 0, x' >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1)
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: x >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs', z = 1 + x + 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: k >= 0, z' = k, z = 0
   goal(z, z') -{ 1 }-> binom(x, y) :|: x >= 0, y >= 0, z = x, z' = y


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

(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:

   @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''')
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1)
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0


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

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

   { @ }
   { binom }
   { goal }

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

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

   @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''')
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1)
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {@}, {binom}, {goal}

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

(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:

   @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''')
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1)
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {@}, {binom}, {goal}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''')
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1)
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {@}, {binom}, {goal}
Previous analysis results are:
  @: runtime: ?, size: O(n^1) [z + z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''')
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1)
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {binom}, {goal}
Previous analysis results are:
  @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

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

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

   @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   @(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   binom(z, z') -{ 5 }-> s' :|: s' >= 0, s' <= 1 + 0 + 0 + (1 + 0 + 0), z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
   binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''')
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1)
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {binom}, {goal}
Previous analysis results are:
  @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: binom
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

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

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

   @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   @(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   binom(z, z') -{ 5 }-> s' :|: s' >= 0, s' <= 1 + 0 + 0 + (1 + 0 + 0), z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
   binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''')
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1)
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {binom}, {goal}
Previous analysis results are:
  @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  binom: runtime: ?, size: EXP

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   @(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   binom(z, z') -{ 5 }-> s' :|: s' >= 0, s' <= 1 + 0 + 0 + (1 + 0 + 0), z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
   binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''')
   binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1)
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0

Function symbols to be analyzed: {binom}, {goal}
Previous analysis results are:
  @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  binom: runtime: INF, size: EXP

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

(31) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

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

(32)
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:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1]
   @(Nil, ys) -> ys [1]
   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) [1]
   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) [1]
   binom(Nil, k) -> Cons(Nil, Nil) [1]
   goal(x, y) -> binom(x, y) [1]

The TRS has the following type information:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
binom :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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

(33) 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

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

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

   @(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0
   @(z, z') -{ 1 }-> 1 + x + @(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0
   binom(z, z') -{ 1 }-> @(binom(xs, xs'), binom(xs, 1 + x' + xs')) :|: z = 1 + x + xs, xs >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: k >= 0, z' = k, z = 0
   goal(z, z') -{ 1 }-> binom(x, y) :|: x >= 0, y >= 0, z = x, z' = y

Only complete derivations are relevant for the runtime complexity.

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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   goal(z0, z1) -> binom(z0, z1)
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   @'(Nil, z0) -> c1
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
   BINOM(Cons(z0, z1), Nil) -> c4
   BINOM(Nil, z0) -> c5
   GOAL(z0, z1) -> c6(BINOM(z0, z1))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   @'(Nil, z0) -> c1
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
   BINOM(Cons(z0, z1), Nil) -> c4
   BINOM(Nil, z0) -> c5
   GOAL(z0, z1) -> c6(BINOM(z0, z1))
K tuples:none
Defined Rule Symbols:   @_2, binom_2, goal_2

Defined Pair Symbols:   @'_2, BINOM_2, GOAL_2

Compound Symbols:   c_1, c1, c2_2, c3_2, c4, c5, c6_1


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

(37) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0, z1) -> c6(BINOM(z0, z1))
Removed 3 trailing nodes:
   BINOM(Nil, z0) -> c5
   @'(Nil, z0) -> c1
   BINOM(Cons(z0, z1), Nil) -> c4

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   goal(z0, z1) -> binom(z0, z1)
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
K tuples:none
Defined Rule Symbols:   @_2, binom_2, goal_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c3_2


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

(39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   goal(z0, z1) -> binom(z0, z1)

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c3_2


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

(41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
We considered the (Usable) Rules:none
And the Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(@(x_1, x_2)) = [1] + x_2
   POL(@'(x_1, x_2)) = 0
   POL(BINOM(x_1, x_2)) = x_1 + x_2
   POL(Cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(Nil) = 0
   POL(binom(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c3_2


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

(43) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) by 
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)), BINOM(Nil, x3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Nil))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))), BINOM(Nil, z0))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)), BINOM(Nil, x3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Nil))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))), BINOM(Nil, z0))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c3_2, c2_2


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

(45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c3_2, c2_2, c2_1


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) by 
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)), BINOM(Nil, Cons(x2, x3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))), BINOM(Nil, Cons(x2, z0)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)), BINOM(Nil, Cons(x2, x3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))), BINOM(Nil, Cons(x2, z0)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c2_1, c3_2


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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c2_1, c3_2, c3_1


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3)) by 
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c2(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))), BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Nil))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c2(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))), BINOM(Cons(x1, Nil), z0))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Nil))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c2(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))), BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Nil))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c2(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))), BINOM(Cons(x1, Nil), z0))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Nil))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c2_1, c3_2, c3_1


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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c2(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))), BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c2(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))), BINOM(Cons(x1, Nil), z0))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c2_1, c3_2, c3_1


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

(55) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c2_1, c3_2, c3_1, c1_1


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) by 
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, Cons(x4, x5))) -> c2(@'(@(binom(z1, x5), binom(z1, Cons(x4, x5))), @(binom(z1, Cons(x4, x5)), binom(z1, Cons(z2, Cons(x4, x5))))), BINOM(Cons(z0, z1), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c2(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))), BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c2(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))), BINOM(Cons(x1, Nil), Cons(x4, z0)))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c2(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))), BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c2(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))), BINOM(Cons(x1, Nil), Cons(x4, z0)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_1, c3_2, c3_1, c2_2, c1_1


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

(59) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_1, c3_2, c3_1, c2_2, c1_1, c4_1


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil))) by 
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_1, c3_2, c3_1, c2_2, c1_1, c4_1


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) by 
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(z2, Nil)))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_1, c3_2, c3_1, c2_2, c1_1, c4_1


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) by 
   BINOM(Cons(x0, Nil), Cons(x1, x2)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c3_2, c3_1, c2_2, c2_1, c1_1, c4_1


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) by 
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c3(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))), BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c3(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))), BINOM(Cons(x1, Nil), Cons(x3, z0)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c3(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))), BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c3(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))), BINOM(Cons(x1, Nil), Cons(x3, z0)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c3_2, c3_1, c2_2, c2_1, c1_1, c4_1


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

(69) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c3_2, c3_1, c2_2, c2_1, c1_1, c4_1, c5_1


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) by 
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, Cons(x4, x5))) -> c3(@'(@(binom(z1, x5), binom(z1, Cons(x4, x5))), @(binom(z1, Cons(x4, x5)), binom(z1, Cons(z2, Cons(x4, x5))))), BINOM(Cons(z0, z1), Cons(z2, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c3(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))), BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c3(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))), BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c3(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))), BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c3(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))), BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c3_2, c3_1, c2_2, c2_1, c1_1, c4_1, c5_1


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

(73) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c3_2, c3_1, c2_2, c2_1, c1_1, c4_1, c5_1, c6_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) by 
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(z2, Nil)))), BINOM(Cons(z0, z1), Cons(z2, Nil)))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c3_1, c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) by 
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c3_1, c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) by 
   BINOM(Cons(x0, Nil), Cons(x1, x2)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1


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

(81) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1


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

(83) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
S tuples:
   @'(Cons(z0, z1), z2) -> c(@'(z1, z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   @'_2, BINOM_2

Compound Symbols:   c_1, c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1


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

(85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) by 
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3))
   BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3)))
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1, c_1


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

(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(z2, Nil)))))
   BINOM(Cons(x0, Nil), Cons(x1, x2)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(BINOM(Cons(x1, Nil), z0))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1


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

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1


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

(91) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1


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

(93) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1


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

(95) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1


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

(97) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1


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

(99) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1


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

(101) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1


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

(103) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1, c4_1


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

(105) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1, c4_1


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

(107) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1, c4_1


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

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1, c4_1


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

(111) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c6_1, c3_1, c2_1, c_1, c1_1, c4_1, c5_1


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

(113) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3)))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c6_1, c3_1, c2_1, c_1, c1_1, c4_1, c5_1


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

(115) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c6_1, c3_1, c2_1, c_1, c1_1, c4_1, c5_1


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

(117) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c6_1, c3_1, c2_1, c_1, c1_1, c4_1, c5_1


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

(119) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c6_1, c3_1, c2_1, c_1, c1_1, c4_1, c5_1


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c2_1, c_1, c1_1, c4_1, c5_1, c6_1


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c2_1, c_1, c1_1, c4_1, c5_1, c6_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c_1, c1_1, c4_1, c5_1, c6_1, c2_1


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

(127) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c_1, c1_1, c4_1, c5_1, c6_1, c2_1


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

(129) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1


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

(131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) by 
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1


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

(133) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1


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

(135) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1


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

(137) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
----------------------------------------

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1


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

(139) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
----------------------------------------

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1


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

(141) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
S tuples:
   @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c_1, c5_1, c6_1, c2_1, c1_1, c4_1


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

(143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) by 
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))
S tuples:
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c5_1, c6_1, c2_1, c1_1, c4_1, c_1


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

(145) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
S tuples:
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c5_1, c6_1, c2_1, c1_1, c4_1, c_1


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

(147) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
S tuples:
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c5_1, c6_1, c2_1, c1_1, c4_1, c_1


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

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
S tuples:
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c5_1, c6_1, c2_1, c1_1, c4_1, c_1


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

(151) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3)))
   binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil)
   binom(Nil, z0) -> Cons(Nil, Nil)
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
Tuples:
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3)))
   BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7)))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4))))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6)))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil))))
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))))
   BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil)))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil)))))
   BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3)))))
S tuples:
   @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3))
K tuples:none
Defined Rule Symbols:   binom_2, @_2

Defined Pair Symbols:   BINOM_2, @'_2

Compound Symbols:   c2_2, c3_2, c3_1, c6_1, c2_1, c1_1, c4_1, c_1, c5_1