KILLED
proof of input_y33eja3eb5.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 203 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 1 ms]
(6) CpxTRS
(7) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
        (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (14) CpxTypedWeightedCompleteTrs
        (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 6 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 47 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 407 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 104 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 672 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 492 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 85 ms]
        (44) CpxRNTS
    (45) CompletionProof [UPPER BOUND(ID), 0 ms]
    (46) CpxTypedWeightedCompleteTrs
        (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
(49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(50) CdtProblem
    (51) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (52) CdtProblem
    (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 195 ms]
    (56) CdtProblem
    (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 17 ms]
    (58) CdtProblem
    (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRhsSimplificationProcessorProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

The (relative) TRS S consists of the following rules:

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

The (relative) TRS S consists of the following rules:

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

The (relative) TRS S consists of the following rules:

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(6)
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:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)
   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(7) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(8)
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:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs)) [1]
   subsets(Nil) -> Cons(Nil, Nil) [1]
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest)) [1]
   mapconsapp(x, Nil, rest) -> rest [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs) -> subsets(xs) [1]
   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs) [0]

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

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs)) [1]
   subsets(Nil) -> Cons(Nil, Nil) [1]
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest)) [1]
   mapconsapp(x, Nil, rest) -> rest [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs) -> subsets(xs) [1]
   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs) [0]

The TRS has the following type information:
subsets :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil

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

(11) 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:

   notEmpty_1
   goal_1

(c) The following functions are completely defined:

   subsets_1
   mapconsapp_3
   subsets[Ite][True][Let]_2

Due to the following rules being added:

   subsets[Ite][True][Let](v0, v1) -> Nil [0]

And the following fresh constants: none

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

(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:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs)) [1]
   subsets(Nil) -> Cons(Nil, Nil) [1]
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest)) [1]
   mapconsapp(x, Nil, rest) -> rest [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs) -> subsets(xs) [1]
   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs) [0]
   subsets[Ite][True][Let](v0, v1) -> Nil [0]

The TRS has the following type information:
subsets :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil

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

(13) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(14)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   subsets(Cons(x, Cons(x'', xs'))) -> subsets[Ite][True][Let](Cons(x, Cons(x'', xs')), subsets[Ite][True][Let](Cons(x'', xs'), subsets(xs'))) [2]
   subsets(Cons(x, Nil)) -> subsets[Ite][True][Let](Cons(x, Nil), Cons(Nil, Nil)) [2]
   subsets(Nil) -> Cons(Nil, Nil) [1]
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest)) [1]
   mapconsapp(x, Nil, rest) -> rest [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs) -> subsets(xs) [1]
   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs) [0]
   subsets[Ite][True][Let](v0, v1) -> Nil [0]

The TRS has the following type information:
subsets :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil

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

(15) 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
   True => 1
   False => 0

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

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

   goal(z) -{ 1 }-> subsets(xs) :|: xs >= 0, z = xs
   mapconsapp(z, z', z'') -{ 1 }-> rest :|: x >= 0, z'' = rest, rest >= 0, z = x, z' = 0
   mapconsapp(z, z', z'') -{ 1 }-> 1 + (1 + x' + x) + mapconsapp(x', xs, rest) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z'' = rest, rest >= 0, z = x'
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + 0, 1 + 0 + 0) :|: x >= 0, z = 1 + x + 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> mapconsapp(x, subs, subs) :|: z = 1 + x + xs, xs >= 0, z' = subs, x >= 0, subs >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 1 }-> 1 + (1 + z + x) + mapconsapp(z, xs, z'') :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> mapconsapp(x, z', z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { notEmpty }
   { mapconsapp }
   { subsets[Ite][True][Let] }
   { subsets }
   { goal }

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 1 }-> 1 + (1 + z + x) + mapconsapp(z, xs, z'') :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> mapconsapp(x, z', z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {notEmpty}, {mapconsapp}, {subsets[Ite][True][Let]}, {subsets}, {goal}

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

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 1 }-> 1 + (1 + z + x) + mapconsapp(z, xs, z'') :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> mapconsapp(x, z', z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {notEmpty}, {mapconsapp}, {subsets[Ite][True][Let]}, {subsets}, {goal}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 1 }-> 1 + (1 + z + x) + mapconsapp(z, xs, z'') :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> mapconsapp(x, z', z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {notEmpty}, {mapconsapp}, {subsets[Ite][True][Let]}, {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: ?, size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 1 }-> 1 + (1 + z + x) + mapconsapp(z, xs, z'') :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> mapconsapp(x, z', z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {mapconsapp}, {subsets[Ite][True][Let]}, {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [1]

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

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 1 }-> 1 + (1 + z + x) + mapconsapp(z, xs, z'') :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> mapconsapp(x, z', z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {mapconsapp}, {subsets[Ite][True][Let]}, {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 1 }-> 1 + (1 + z + x) + mapconsapp(z, xs, z'') :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> mapconsapp(x, z', z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {mapconsapp}, {subsets[Ite][True][Let]}, {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  mapconsapp: runtime: ?, size: O(n^2) [z*z' + z' + z'^2 + z'']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 1 }-> 1 + (1 + z + x) + mapconsapp(z, xs, z'') :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> mapconsapp(x, z', z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {subsets[Ite][True][Let]}, {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  mapconsapp: runtime: O(n^1) [1 + z'], size: O(n^2) [z*z' + z' + z'^2 + z'']

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

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 2 + xs }-> 1 + (1 + z + x) + s :|: s >= 0, s <= xs * z + xs * xs + xs + z'', xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z' * x + z' * z' + z' + z', z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {subsets[Ite][True][Let]}, {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  mapconsapp: runtime: O(n^1) [1 + z'], size: O(n^2) [z*z' + z' + z'^2 + z'']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: subsets[Ite][True][Let]
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 2 + xs }-> 1 + (1 + z + x) + s :|: s >= 0, s <= xs * z + xs * xs + xs + z'', xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z' * x + z' * z' + z' + z', z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {subsets[Ite][True][Let]}, {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  mapconsapp: runtime: O(n^1) [1 + z'], size: O(n^2) [z*z' + z' + z'^2 + z'']
  subsets[Ite][True][Let]: runtime: ?, size: INF

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: subsets[Ite][True][Let]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 2 + xs }-> 1 + (1 + z + x) + s :|: s >= 0, s <= xs * z + xs * xs + xs + z'', xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + (z - 1) + 0, 1 + 0 + 0) :|: z - 1 >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z' * x + z' * z' + z' + z', z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  mapconsapp: runtime: O(n^1) [1 + z'], size: O(n^2) [z*z' + z' + z'^2 + z'']
  subsets[Ite][True][Let]: runtime: O(n^1) [1 + z'], size: INF

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

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 2 + xs }-> 1 + (1 + z + x) + s :|: s >= 0, s <= xs * z + xs * xs + xs + z'', xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= inf, z - 1 >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z' * x + z' * z' + z' + z', z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  mapconsapp: runtime: O(n^1) [1 + z'], size: O(n^2) [z*z' + z' + z'^2 + z'']
  subsets[Ite][True][Let]: runtime: O(n^1) [1 + z'], size: INF

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 2 + xs }-> 1 + (1 + z + x) + s :|: s >= 0, s <= xs * z + xs * xs + xs + z'', xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= inf, z - 1 >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z' * x + z' * z' + z' + z', z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  mapconsapp: runtime: O(n^1) [1 + z'], size: O(n^2) [z*z' + z' + z'^2 + z'']
  subsets[Ite][True][Let]: runtime: O(n^1) [1 + z'], size: INF
  subsets: runtime: ?, size: INF

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z) -{ 1 }-> subsets(z) :|: z >= 0
   mapconsapp(z, z', z'') -{ 1 }-> z'' :|: z >= 0, z'' >= 0, z' = 0
   mapconsapp(z, z', z'') -{ 2 + xs }-> 1 + (1 + z + x) + s :|: s >= 0, s <= xs * z + xs * xs + xs + z'', xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   subsets(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= inf, z - 1 >= 0
   subsets(z) -{ 2 }-> subsets[Ite][True][Let](1 + x + (1 + x'' + xs'), subsets[Ite][True][Let](1 + x'' + xs', subsets(xs'))) :|: z = 1 + x + (1 + x'' + xs'), x >= 0, xs' >= 0, x'' >= 0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z' * x + z' * z' + z' + z', z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {subsets}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  mapconsapp: runtime: O(n^1) [1 + z'], size: O(n^2) [z*z' + z' + z'^2 + z'']
  subsets[Ite][True][Let]: runtime: O(n^1) [1 + z'], size: INF
  subsets: runtime: INF, size: INF

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

(45) 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:

   subsets[Ite][True][Let](v0, v1) -> null_subsets[Ite][True][Let] [0]
   subsets(v0) -> null_subsets [0]
   mapconsapp(v0, v1, v2) -> null_mapconsapp [0]
   notEmpty(v0) -> null_notEmpty [0]

And the following fresh constants:    null_subsets[Ite][True][Let], null_subsets, null_mapconsapp, null_notEmpty

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

(46)
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:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs)) [1]
   subsets(Nil) -> Cons(Nil, Nil) [1]
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest)) [1]
   mapconsapp(x, Nil, rest) -> rest [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs) -> subsets(xs) [1]
   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs) [0]
   subsets[Ite][True][Let](v0, v1) -> null_subsets[Ite][True][Let] [0]
   subsets(v0) -> null_subsets [0]
   mapconsapp(v0, v1, v2) -> null_mapconsapp [0]
   notEmpty(v0) -> null_notEmpty [0]

The TRS has the following type information:
subsets :: Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp -> Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp
Cons :: Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp -> Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp -> Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp
subsets[Ite][True][Let] :: Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp -> Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp -> Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp
Nil :: Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp
mapconsapp :: Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp -> Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp -> Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp -> Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp
notEmpty :: Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp -> True:False:null_notEmpty
True :: True:False:null_notEmpty
False :: True:False:null_notEmpty
goal :: Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp -> Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp
null_subsets[Ite][True][Let] :: Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp
null_subsets :: Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp
null_mapconsapp :: Cons:Nil:null_subsets[Ite][True][Let]:null_subsets:null_mapconsapp
null_notEmpty :: True:False:null_notEmpty

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

(47) 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
   True => 2
   False => 1
   null_subsets[Ite][True][Let] => 0
   null_subsets => 0
   null_mapconsapp => 0
   null_notEmpty => 0

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

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

   goal(z) -{ 1 }-> subsets(xs) :|: xs >= 0, z = xs
   mapconsapp(z, z', z'') -{ 1 }-> rest :|: x >= 0, z'' = rest, rest >= 0, z = x, z' = 0
   mapconsapp(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   mapconsapp(z, z', z'') -{ 1 }-> 1 + (1 + x' + x) + mapconsapp(x', xs, rest) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z'' = rest, rest >= 0, z = x'
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   notEmpty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   subsets(z) -{ 1 }-> subsets[Ite][True][Let](1 + x + xs, subsets(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   subsets(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   subsets(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> mapconsapp(x, subs, subs) :|: z = 1 + x + xs, xs >= 0, z' = subs, x >= 0, subs >= 0
   subsets[Ite][True][Let](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> subsets(z0)
Tuples:
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   SUBSETS(Nil) -> c2
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   GOAL(z0) -> c7(SUBSETS(z0))
S tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   SUBSETS(Nil) -> c2
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   GOAL(z0) -> c7(SUBSETS(z0))
K tuples:none
Defined Rule Symbols:   subsets_1, mapconsapp_3, notEmpty_1, goal_1, subsets[Ite][True][Let]_2

Defined Pair Symbols:   SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1, MAPCONSAPP_3, NOTEMPTY_1, GOAL_1

Compound Symbols:   c_1, c1_2, c2, c3_1, c4, c5, c6, c7_1


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

(51) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0) -> c7(SUBSETS(z0))
Removed 3 trailing nodes:
   NOTEMPTY(Nil) -> c6
   SUBSETS(Nil) -> c2
   NOTEMPTY(Cons(z0, z1)) -> c5

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> subsets(z0)
Tuples:
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
S tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
K tuples:none
Defined Rule Symbols:   subsets_1, mapconsapp_3, notEmpty_1, goal_1, subsets[Ite][True][Let]_2

Defined Pair Symbols:   SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1, MAPCONSAPP_3

Compound Symbols:   c_1, c1_2, c3_1, c4


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

(53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> subsets(z0)

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
S tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
K tuples:none
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1, MAPCONSAPP_3

Compound Symbols:   c_1, c1_2, c3_1, c4


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

(55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MAPCONSAPP(z0, Nil, z1) -> c4
We considered the (Usable) Rules:none
And the Tuples:
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(MAPCONSAPP(x_1, x_2, x_3)) = [1]
   POL(Nil) = 0
   POL(SUBSETS(x_1)) = x_1 + x_1^3
   POL(SUBSETS[ITE][TRUE][LET](x_1, x_2)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4) = 0
   POL(mapconsapp(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 + x_3^2 + x_2*x_3 + x_1*x_3 + x_1^2 + x_1*x_2 + x_2^2 + x_2^3 + x_1*x_2^2 + x_2^2*x_3 + x_2*x_3^2 + x_1*x_2*x_3 + x_1^2*x_2 + x_1^3 + x_1^2*x_3 + x_1*x_3^2 + x_3^3
   POL(subsets(x_1)) = 0
   POL(subsets[Ite][True][Let](x_1, x_2)) = [1] + x_1 + x_1^2 + x_1^3 + x_1^2*x_2 + x_1*x_2^2 + x_2^3

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
S tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   MAPCONSAPP(z0, Nil, z1) -> c4
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1, MAPCONSAPP_3

Compound Symbols:   c_1, c1_2, c3_1, c4


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

(57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
We considered the (Usable) Rules:none
And the Tuples:
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(MAPCONSAPP(x_1, x_2, x_3)) = 0
   POL(Nil) = 0
   POL(SUBSETS(x_1)) = x_1
   POL(SUBSETS[ITE][TRUE][LET](x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4) = 0
   POL(mapconsapp(x_1, x_2, x_3)) = [3] + [3]x_1 + [3]x_3
   POL(subsets(x_1)) = [3]
   POL(subsets[Ite][True][Let](x_1, x_2)) = [3]

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   MAPCONSAPP(z0, Nil, z1) -> c4
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1, MAPCONSAPP_3

Compound Symbols:   c_1, c1_2, c3_1, c4


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

(59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2)) by 
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Nil) -> c(MAPCONSAPP(z0, Nil, Nil))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Nil) -> c(MAPCONSAPP(z0, Nil, Nil))
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   MAPCONSAPP(z0, Nil, z1) -> c4
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS_1, MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2

Compound Symbols:   c1_2, c3_1, c4, c_1


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

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Nil) -> c(MAPCONSAPP(z0, Nil, Nil))
   MAPCONSAPP(z0, Nil, z1) -> c4

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS_1, MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2

Compound Symbols:   c1_2, c3_1, c_1


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1)) by 
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)), SUBSETS(Nil))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)), SUBSETS(Nil))
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1

Compound Symbols:   c3_1, c_1, c1_2


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

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)))
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1

Compound Symbols:   c3_1, c_1, c1_2, c1_1


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))), SUBSETS(Cons(z0, z1))) by 
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Cons(x1, Cons(z0, z1)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Cons(z0, z1))), subsets[Ite][True][Let](Cons(x1, Cons(z0, z1)), subsets[Ite][True][Let](Cons(z0, z1), subsets(z1)))), SUBSETS(Cons(x1, Cons(z0, z1))))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), subsets[Ite][True][Let](Cons(x1, Nil), Cons(Nil, Nil))), SUBSETS(Cons(x1, Nil)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)))
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Cons(x1, Cons(z0, z1)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Cons(z0, z1))), subsets[Ite][True][Let](Cons(x1, Cons(z0, z1)), subsets[Ite][True][Let](Cons(z0, z1), subsets(z1)))), SUBSETS(Cons(x1, Cons(z0, z1))))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), subsets[Ite][True][Let](Cons(x1, Nil), Cons(Nil, Nil))), SUBSETS(Cons(x1, Nil)))
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1

Compound Symbols:   c3_1, c_1, c1_1, c1_2


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

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)))
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Cons(x1, Cons(z0, z1)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Cons(z0, z1))), subsets[Ite][True][Let](Cons(x1, Cons(z0, z1)), subsets[Ite][True][Let](Cons(z0, z1), subsets(z1)))), SUBSETS(Cons(x1, Cons(z0, z1))))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), subsets[Ite][True][Let](Cons(x1, Nil), Cons(Nil, Nil))))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS(Cons(x1, Nil)))
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1

Compound Symbols:   c3_1, c_1, c1_1, c1_2, c2_1


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

(71) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUBSETS(Cons(x0, Cons(x1, Cons(z0, z1)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Cons(z0, z1))), subsets[Ite][True][Let](Cons(x1, Cons(z0, z1)), subsets[Ite][True][Let](Cons(z0, z1), subsets(z1)))), SUBSETS(Cons(x1, Cons(z0, z1)))) by SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
----------------------------------------

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)))
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), subsets[Ite][True][Let](Cons(x1, Nil), Cons(Nil, Nil))))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS(Cons(x1, Nil)))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1

Compound Symbols:   c3_1, c_1, c1_1, c1_2, c2_1


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

(73) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), subsets[Ite][True][Let](Cons(x1, Nil), Cons(Nil, Nil)))) by SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), mapconsapp(z1, Cons(Nil, Nil), Cons(Nil, Nil))))
----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)))
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS(Cons(x1, Nil)))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
   SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), mapconsapp(z1, Cons(Nil, Nil), Cons(Nil, Nil))))
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1

Compound Symbols:   c3_1, c_1, c1_1, c1_2, c2_1


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

(75) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3)))) by SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), mapconsapp(z2, subsets(z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)))
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS(Cons(x1, Nil)))
   SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), mapconsapp(z1, Cons(Nil, Nil), Cons(Nil, Nil))))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), mapconsapp(z2, subsets(z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1

Compound Symbols:   c3_1, c_1, c1_1, c1_2, c2_1


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

(77) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), mapconsapp(z1, Cons(Nil, Nil), Cons(Nil, Nil)))) by SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), mapconsapp(z1, Nil, Cons(Nil, Nil)))))
----------------------------------------

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)))
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS(Cons(x1, Nil)))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), mapconsapp(z2, subsets(z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
   SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), mapconsapp(z1, Nil, Cons(Nil, Nil)))))
S tuples:
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1

Compound Symbols:   c3_1, c_1, c1_1, c1_2, c2_1


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

(79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3)) by 
   MAPCONSAPP(z0, Cons(z1, Cons(y1, y2)), z3) -> c3(MAPCONSAPP(z0, Cons(y1, y2), z3))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)))
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS(Cons(x1, Nil)))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), mapconsapp(z2, subsets(z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
   SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), mapconsapp(z1, Nil, Cons(Nil, Nil)))))
   MAPCONSAPP(z0, Cons(z1, Cons(y1, y2)), z3) -> c3(MAPCONSAPP(z0, Cons(y1, y2), z3))
S tuples:
   MAPCONSAPP(z0, Cons(z1, Cons(y1, y2)), z3) -> c3(MAPCONSAPP(z0, Cons(y1, y2), z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1, MAPCONSAPP_3

Compound Symbols:   c_1, c1_1, c1_2, c2_1, c3_1


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

(81) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace SUBSETS[ITE][TRUE][LET](Cons(z0, z1), Cons(y1, y2)) -> c(MAPCONSAPP(z0, Cons(y1, y2), Cons(y1, y2))) by 
   SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)) -> c(MAPCONSAPP(x0, Cons(Nil, Nil), Cons(Nil, Nil)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Cons(x2, x3))), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), Cons(Cons(x1, Nil), y0)) -> c(MAPCONSAPP(x0, Cons(Cons(x1, Nil), y0), Cons(Cons(x1, Nil), y0)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)))
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS(Cons(x1, Nil)))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), mapconsapp(z2, subsets(z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
   SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), mapconsapp(z1, Nil, Cons(Nil, Nil)))))
   MAPCONSAPP(z0, Cons(z1, Cons(y1, y2)), z3) -> c3(MAPCONSAPP(z0, Cons(y1, y2), z3))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)) -> c(MAPCONSAPP(x0, Cons(Nil, Nil), Cons(Nil, Nil)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Cons(x2, x3))), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), Cons(Cons(x1, Nil), y0)) -> c(MAPCONSAPP(x0, Cons(Cons(x1, Nil), y0), Cons(Cons(x1, Nil), y0)))
S tuples:
   MAPCONSAPP(z0, Cons(z1, Cons(y1, y2)), z3) -> c3(MAPCONSAPP(z0, Cons(y1, y2), z3))
K tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS_1, MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2

Compound Symbols:   c1_1, c1_2, c2_1, c3_1, c_1


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

(83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)) -> c(MAPCONSAPP(x0, Cons(Nil, Nil), Cons(Nil, Nil)))
   SUBSETS(Cons(x0, Cons(x1, Nil))) -> c2(SUBSETS(Cons(x1, Nil)))
   SUBSETS(Cons(x0, Nil)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Nil), Cons(Nil, Nil)))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), mapconsapp(z2, subsets(z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
   SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), mapconsapp(z1, Nil, Cons(Nil, Nil)))))
   MAPCONSAPP(z0, Cons(z1, Cons(y1, y2)), z3) -> c3(MAPCONSAPP(z0, Cons(y1, y2), z3))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Cons(x2, x3))), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), Cons(Cons(x1, Nil), y0)) -> c(MAPCONSAPP(x0, Cons(Cons(x1, Nil), y0), Cons(Cons(x1, Nil), y0)))
S tuples:
   MAPCONSAPP(z0, Cons(z1, Cons(y1, y2)), z3) -> c3(MAPCONSAPP(z0, Cons(y1, y2), z3))
K tuples:none
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS_1, MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2

Compound Symbols:   c1_2, c2_1, c3_1, c_1


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

(85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MAPCONSAPP(z0, Cons(z1, Cons(y1, y2)), z3) -> c3(MAPCONSAPP(z0, Cons(y1, y2), z3)) by 
   MAPCONSAPP(z0, Cons(z1, Cons(z2, Cons(y2, y3))), z4) -> c3(MAPCONSAPP(z0, Cons(z2, Cons(y2, y3)), z4))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), mapconsapp(z2, subsets(z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
   SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), mapconsapp(z1, Nil, Cons(Nil, Nil)))))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Cons(x2, x3))), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), Cons(Cons(x1, Nil), y0)) -> c(MAPCONSAPP(x0, Cons(Cons(x1, Nil), y0), Cons(Cons(x1, Nil), y0)))
   MAPCONSAPP(z0, Cons(z1, Cons(z2, Cons(y2, y3))), z4) -> c3(MAPCONSAPP(z0, Cons(z2, Cons(y2, y3)), z4))
S tuples:
   MAPCONSAPP(z0, Cons(z1, Cons(z2, Cons(y2, y3))), z4) -> c3(MAPCONSAPP(z0, Cons(z2, Cons(y2, y3)), z4))
K tuples:none
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS_1, SUBSETS[ITE][TRUE][LET]_2, MAPCONSAPP_3

Compound Symbols:   c1_2, c2_1, c_1, c3_1


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

(87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3))) by 
   SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, z2)), Cons(z3, Cons(y2, Cons(y3, y4)))) -> c(MAPCONSAPP(z0, Cons(z3, Cons(y2, Cons(y3, y4))), Cons(z3, Cons(y2, Cons(y3, y4)))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), mapconsapp(z2, subsets(z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
   SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), mapconsapp(z1, Nil, Cons(Nil, Nil)))))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Cons(x2, x3))), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3)))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), Cons(Cons(x1, Nil), y0)) -> c(MAPCONSAPP(x0, Cons(Cons(x1, Nil), y0), Cons(Cons(x1, Nil), y0)))
   MAPCONSAPP(z0, Cons(z1, Cons(z2, Cons(y2, y3))), z4) -> c3(MAPCONSAPP(z0, Cons(z2, Cons(y2, y3)), z4))
   SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, z2)), Cons(z3, Cons(y2, Cons(y3, y4)))) -> c(MAPCONSAPP(z0, Cons(z3, Cons(y2, Cons(y3, y4))), Cons(z3, Cons(y2, Cons(y3, y4)))))
S tuples:
   MAPCONSAPP(z0, Cons(z1, Cons(z2, Cons(y2, y3))), z4) -> c3(MAPCONSAPP(z0, Cons(z2, Cons(y2, y3)), z4))
K tuples:none
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS_1, SUBSETS[ITE][TRUE][LET]_2, MAPCONSAPP_3

Compound Symbols:   c1_2, c2_1, c_1, c3_1


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

(89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Cons(x2, x3))), Cons(z2, z3)) -> c(MAPCONSAPP(x0, Cons(z2, z3), Cons(z2, z3))) by 
   SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(y2, Cons(y3, y4)))) -> c(MAPCONSAPP(z0, Cons(z4, Cons(y2, Cons(y3, y4))), Cons(z4, Cons(y2, Cons(y3, y4)))))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), mapconsapp(z2, subsets(z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
   SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), mapconsapp(z1, Nil, Cons(Nil, Nil)))))
   SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), Cons(Cons(x1, Nil), y0)) -> c(MAPCONSAPP(x0, Cons(Cons(x1, Nil), y0), Cons(Cons(x1, Nil), y0)))
   MAPCONSAPP(z0, Cons(z1, Cons(z2, Cons(y2, y3))), z4) -> c3(MAPCONSAPP(z0, Cons(z2, Cons(y2, y3)), z4))
   SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, z2)), Cons(z3, Cons(y2, Cons(y3, y4)))) -> c(MAPCONSAPP(z0, Cons(z3, Cons(y2, Cons(y3, y4))), Cons(z3, Cons(y2, Cons(y3, y4)))))
   SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(y2, Cons(y3, y4)))) -> c(MAPCONSAPP(z0, Cons(z4, Cons(y2, Cons(y3, y4))), Cons(z4, Cons(y2, Cons(y3, y4)))))
S tuples:
   MAPCONSAPP(z0, Cons(z1, Cons(z2, Cons(y2, y3))), z4) -> c3(MAPCONSAPP(z0, Cons(z2, Cons(y2, y3)), z4))
K tuples:none
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS_1, SUBSETS[ITE][TRUE][LET]_2, MAPCONSAPP_3

Compound Symbols:   c1_2, c2_1, c_1, c3_1


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

(91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(x1, Nil)), Cons(Cons(x1, Nil), y0)) -> c(MAPCONSAPP(x0, Cons(Cons(x1, Nil), y0), Cons(Cons(x1, Nil), y0))) by 
   SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), Cons(y2, Cons(y3, y4)))) -> c(MAPCONSAPP(z0, Cons(Cons(z1, Nil), Cons(y2, Cons(y3, y4))), Cons(Cons(z1, Nil), Cons(y2, Cons(y3, y4)))))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
Tuples:
   SUBSETS(Cons(x0, Cons(z0, z1))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(x0, Cons(z0, z1)), mapconsapp(z0, subsets(z1), subsets(z1))), SUBSETS(Cons(z0, z1)))
   SUBSETS(Cons(z0, Cons(z1, Cons(z2, z3)))) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), mapconsapp(z1, subsets[Ite][True][Let](Cons(z2, z3), subsets(z3)), mapconsapp(z2, subsets(z3), subsets(z3)))), SUBSETS(Cons(z1, Cons(z2, z3))))
   SUBSETS(Cons(z0, Cons(z1, Nil))) -> c2(SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), mapconsapp(z1, Nil, Cons(Nil, Nil)))))
   MAPCONSAPP(z0, Cons(z1, Cons(z2, Cons(y2, y3))), z4) -> c3(MAPCONSAPP(z0, Cons(z2, Cons(y2, y3)), z4))
   SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, z2)), Cons(z3, Cons(y2, Cons(y3, y4)))) -> c(MAPCONSAPP(z0, Cons(z3, Cons(y2, Cons(y3, y4))), Cons(z3, Cons(y2, Cons(y3, y4)))))
   SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(y2, Cons(y3, y4)))) -> c(MAPCONSAPP(z0, Cons(z4, Cons(y2, Cons(y3, y4))), Cons(z4, Cons(y2, Cons(y3, y4)))))
   SUBSETS[ITE][TRUE][LET](Cons(z0, Cons(z1, Nil)), Cons(Cons(z1, Nil), Cons(y2, Cons(y3, y4)))) -> c(MAPCONSAPP(z0, Cons(Cons(z1, Nil), Cons(y2, Cons(y3, y4))), Cons(Cons(z1, Nil), Cons(y2, Cons(y3, y4)))))
S tuples:
   MAPCONSAPP(z0, Cons(z1, Cons(z2, Cons(y2, y3))), z4) -> c3(MAPCONSAPP(z0, Cons(z2, Cons(y2, y3)), z4))
K tuples:none
Defined Rule Symbols:   subsets_1, subsets[Ite][True][Let]_2, mapconsapp_3

Defined Pair Symbols:   SUBSETS_1, MAPCONSAPP_3, SUBSETS[ITE][TRUE][LET]_2

Compound Symbols:   c1_2, c2_1, c3_1, c_1