KILLED
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 222 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) typed CpxTrs
    (9) OrderProof [LOWER BOUND(ID), 0 ms]
    (10) typed CpxTrs
    (11) RewriteLemmaProof [LOWER BOUND(ID), 7259 ms]
    (12) BOUNDS(1, INF)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(18) CpxRelTRS
    (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxWeightedTrs
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxTypedWeightedTrs
    (23) CompletionProof [UPPER BOUND(ID), 0 ms]
    (24) CpxTypedWeightedCompleteTrs
        (25) NarrowingProof [BOTH BOUNDS(ID, ID), 9 ms]
        (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 9 ms]
        (28) CpxRNTS
        (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 11 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 1 ms]
    (44) CdtProblem
    (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 23 ms]
    (48) CdtProblem
    (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms]
    (54) CdtProblem
    (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 7 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 79 ms]
    (66) CdtProblem
    (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 22 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms]
    (86) CdtProblem
    (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   app(app(ack, 0), y) -> app(succ, y)
   app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
   app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(app(filter, f), nil) -> nil
   app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs)
   app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs))
   app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs)

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false


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

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


The TRS R consists of the following rules:

   app(app(ack, 0), y) -> app(succ, y)
   app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
   app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(app(filter, f), nil) -> nil
   app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs)
   app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs))
   app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs)

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

   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false

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

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

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


The TRS R consists of the following rules:

   app(app(ack, 0), y) -> app(succ, y)
   app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
   app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(app(filter, f), nil) -> nil
   app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs)
   app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs))
   app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs)

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

   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   app(app(ack, 0'), y) -> app(succ, y)
   app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0'))
   app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(app(filter, f), nil) -> nil
   app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs)
   app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs))
   app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs)

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

   encArg(ack) -> ack
   encArg(0') -> 0'
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_ack -> ack
   encode_0 -> 0'
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false

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

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

(8)
Obligation:
Innermost TRS:
Rules:
app(app(ack, 0'), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0'))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs)
encArg(ack) -> ack
encArg(0') -> 0'
encArg(succ) -> succ
encArg(map) -> map
encArg(nil) -> nil
encArg(cons) -> cons
encArg(filter) -> filter
encArg(filter2) -> filter2
encArg(true) -> true
encArg(false) -> false
encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
encode_ack -> ack
encode_0 -> 0'
encode_succ -> succ
encode_map -> map
encode_nil -> nil
encode_cons -> cons
encode_filter -> filter
encode_filter2 -> filter2
encode_true -> true
encode_false -> false

Types:
app :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
ack :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
0' :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
succ :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
map :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
nil :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
cons :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
filter :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
filter2 :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
true :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
false :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encArg :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
cons_app :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_app :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_ack :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_0 :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_succ :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_map :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_nil :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_cons :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_filter :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_filter2 :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_true :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_false :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
hole_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app1_3 :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
gen_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app2_3 :: Nat -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
app, encArg

They will be analysed ascendingly in the following order:
app < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
app(app(ack, 0'), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0'))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs)
encArg(ack) -> ack
encArg(0') -> 0'
encArg(succ) -> succ
encArg(map) -> map
encArg(nil) -> nil
encArg(cons) -> cons
encArg(filter) -> filter
encArg(filter2) -> filter2
encArg(true) -> true
encArg(false) -> false
encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
encode_ack -> ack
encode_0 -> 0'
encode_succ -> succ
encode_map -> map
encode_nil -> nil
encode_cons -> cons
encode_filter -> filter
encode_filter2 -> filter2
encode_true -> true
encode_false -> false

Types:
app :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
ack :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
0' :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
succ :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
map :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
nil :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
cons :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
filter :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
filter2 :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
true :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
false :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encArg :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
cons_app :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_app :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_ack :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_0 :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_succ :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_map :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_nil :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_cons :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_filter :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_filter2 :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_true :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
encode_false :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
hole_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app1_3 :: ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app
gen_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app2_3 :: Nat -> ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app


Generator Equations:
gen_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app2_3(0) <=> ack
gen_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app2_3(+(x, 1)) <=> cons_app(ack, gen_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app2_3(x))


The following defined symbols remain to be analysed:
app, encArg

They will be analysed ascendingly in the following order:
app < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app2_3(n3571_3)) -> *3_3, rt in Omega(0)

Induction Base:
encArg(gen_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app2_3(0))

Induction Step:
encArg(gen_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app2_3(+(n3571_3, 1))) ->_R^Omega(0)
app(encArg(ack), encArg(gen_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app2_3(n3571_3))) ->_R^Omega(0)
app(ack, encArg(gen_ack:0':succ:map:nil:cons:filter:filter2:true:false:cons_app2_3(n3571_3))) ->_IH
app(ack, *3_3)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(12)
BOUNDS(1, INF)

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

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   app(app(ack, 0), y) -> app(succ, y)
   app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
   app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(app(filter, f), nil) -> nil
   app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs)
   app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs))
   app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs)

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

   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false

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

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

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   app(app(ack, 0), y) -> app(succ, y)
   app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
   app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(app(filter, f), nil) -> nil
   app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs)
   app(app(app(app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs))
   app(app(app(app(filter2, false), f), x), xs) -> app(app(filter, f), xs)
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false

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

(17) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs))
   app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(app(filter, f), xs)
   app(c_app(map, f), nil) -> nil
   app(c_app(filter, f), nil) -> nil
   app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs)
   app(c_app(ack, c_app(succ, x)), c_app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))
   app(c_app(ack, c_app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
   app(c_app(ack, 0), y) -> app(succ, y)

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

   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false
   app(x0, x1) -> c_app(x0, x1)

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

(19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(20)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(app(filter, f), xs) [1]
   app(c_app(map, f), nil) -> nil [1]
   app(c_app(filter, f), nil) -> nil [1]
   app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) [1]
   app(c_app(ack, c_app(succ, x)), c_app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y)) [1]
   app(c_app(ack, c_app(succ, x)), y) -> app(app(ack, x), app(succ, 0)) [1]
   app(c_app(ack, 0), y) -> app(succ, y) [1]
   encArg(ack) -> ack [0]
   encArg(0) -> 0 [0]
   encArg(succ) -> succ [0]
   encArg(map) -> map [0]
   encArg(nil) -> nil [0]
   encArg(cons) -> cons [0]
   encArg(filter) -> filter [0]
   encArg(filter2) -> filter2 [0]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0]
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0]
   encode_ack -> ack [0]
   encode_0 -> 0 [0]
   encode_succ -> succ [0]
   encode_map -> map [0]
   encode_nil -> nil [0]
   encode_cons -> cons [0]
   encode_filter -> filter [0]
   encode_filter2 -> filter2 [0]
   encode_true -> true [0]
   encode_false -> false [0]
   app(x0, x1) -> c_app(x0, x1) [0]

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

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

(22)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(app(filter, f), xs) [1]
   app(c_app(map, f), nil) -> nil [1]
   app(c_app(filter, f), nil) -> nil [1]
   app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) [1]
   app(c_app(ack, c_app(succ, x)), c_app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y)) [1]
   app(c_app(ack, c_app(succ, x)), y) -> app(app(ack, x), app(succ, 0)) [1]
   app(c_app(ack, 0), y) -> app(succ, y) [1]
   encArg(ack) -> ack [0]
   encArg(0) -> 0 [0]
   encArg(succ) -> succ [0]
   encArg(map) -> map [0]
   encArg(nil) -> nil [0]
   encArg(cons) -> cons [0]
   encArg(filter) -> filter [0]
   encArg(filter2) -> filter2 [0]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0]
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0]
   encode_ack -> ack [0]
   encode_0 -> 0 [0]
   encode_succ -> succ [0]
   encode_map -> map [0]
   encode_nil -> nil [0]
   encode_cons -> cons [0]
   encode_filter -> filter [0]
   encode_filter2 -> filter2 [0]
   encode_true -> true [0]
   encode_false -> false [0]
   app(x0, x1) -> c_app(x0, x1) [0]

The TRS has the following type information:
app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
c_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encArg :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
cons_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app
encode_false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app

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

(23) 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:
none

(c) The following functions are completely defined:

   encArg_1
   encode_app_2
   encode_ack
   encode_0
   encode_succ
   encode_map
   encode_nil
   encode_cons
   encode_filter
   encode_filter2
   encode_true
   encode_false
   app_2

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_app(v0, v1) -> null_encode_app [0]
   encode_ack -> null_encode_ack [0]
   encode_0 -> null_encode_0 [0]
   encode_succ -> null_encode_succ [0]
   encode_map -> null_encode_map [0]
   encode_nil -> null_encode_nil [0]
   encode_cons -> null_encode_cons [0]
   encode_filter -> null_encode_filter [0]
   encode_filter2 -> null_encode_filter2 [0]
   encode_true -> null_encode_true [0]
   encode_false -> null_encode_false [0]
   app(v0, v1) -> null_app [0]

And the following fresh constants:    null_encArg, null_encode_app, null_encode_ack, null_encode_0, null_encode_succ, null_encode_map, null_encode_nil, null_encode_cons, null_encode_filter, null_encode_filter2, null_encode_true, null_encode_false, null_app

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

(24)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(app(filter, f), xs) [1]
   app(c_app(map, f), nil) -> nil [1]
   app(c_app(filter, f), nil) -> nil [1]
   app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) [1]
   app(c_app(ack, c_app(succ, x)), c_app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y)) [1]
   app(c_app(ack, c_app(succ, x)), y) -> app(app(ack, x), app(succ, 0)) [1]
   app(c_app(ack, 0), y) -> app(succ, y) [1]
   encArg(ack) -> ack [0]
   encArg(0) -> 0 [0]
   encArg(succ) -> succ [0]
   encArg(map) -> map [0]
   encArg(nil) -> nil [0]
   encArg(cons) -> cons [0]
   encArg(filter) -> filter [0]
   encArg(filter2) -> filter2 [0]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0]
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0]
   encode_ack -> ack [0]
   encode_0 -> 0 [0]
   encode_succ -> succ [0]
   encode_map -> map [0]
   encode_nil -> nil [0]
   encode_cons -> cons [0]
   encode_filter -> filter [0]
   encode_filter2 -> filter2 [0]
   encode_true -> true [0]
   encode_false -> false [0]
   app(x0, x1) -> c_app(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_app(v0, v1) -> null_encode_app [0]
   encode_ack -> null_encode_ack [0]
   encode_0 -> null_encode_0 [0]
   encode_succ -> null_encode_succ [0]
   encode_map -> null_encode_map [0]
   encode_nil -> null_encode_nil [0]
   encode_cons -> null_encode_cons [0]
   encode_filter -> null_encode_filter [0]
   encode_filter2 -> null_encode_filter2 [0]
   encode_true -> null_encode_true [0]
   encode_false -> null_encode_false [0]
   app(v0, v1) -> null_app [0]

The TRS has the following type information:
app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
c_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encArg :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
cons_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encArg :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app

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

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

(26)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(c_app(cons, x), app(c_app(filter, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(c_app(cons, x), app(null_app, xs)) [1]
   app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(null_app, app(c_app(filter, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(null_app, app(null_app, xs)) [1]
   app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(c_app(filter, f), xs) [1]
   app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(null_app, xs) [1]
   app(c_app(map, f), nil) -> nil [1]
   app(c_app(filter, f), nil) -> nil [1]
   app(c_app(filter, c_app(map, f5)), c_app(c_app(cons, c_app(c_app(cons, x6), xs1)), xs)) -> app(app(app(app(filter2, app(app(cons, app(f5, x6)), app(app(map, f5), xs1))), c_app(map, f5)), c_app(c_app(cons, x6), xs1)), xs) [2]
   app(c_app(filter, c_app(c_app(c_app(filter2, true), f6), x7)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(cons, x7), app(app(filter, f6), x))), c_app(c_app(c_app(filter2, true), f6), x7)), x), xs) [2]
   app(c_app(filter, c_app(c_app(c_app(filter2, false), f7), x8)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(filter, f7), x)), c_app(c_app(c_app(filter2, false), f7), x8)), x), xs) [2]
   app(c_app(filter, c_app(map, f8)), c_app(c_app(cons, nil), xs)) -> app(app(app(app(filter2, nil), c_app(map, f8)), nil), xs) [2]
   app(c_app(filter, c_app(filter, f9)), c_app(c_app(cons, nil), xs)) -> app(app(app(app(filter2, nil), c_app(filter, f9)), nil), xs) [2]
   app(c_app(filter, c_app(filter, f10)), c_app(c_app(cons, c_app(c_app(cons, x9), xs2)), xs)) -> app(app(app(app(filter2, app(app(app(app(filter2, app(f10, x9)), f10), x9), xs2)), c_app(filter, f10)), c_app(c_app(cons, x9), xs2)), xs) [2]
   app(c_app(filter, c_app(ack, c_app(succ, x10))), c_app(c_app(cons, c_app(succ, y'')), xs)) -> app(app(app(app(filter2, app(app(ack, x10), app(app(ack, app(succ, x10)), y''))), c_app(ack, c_app(succ, x10))), c_app(succ, y'')), xs) [2]
   app(c_app(filter, c_app(ack, c_app(succ, x11))), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(ack, x11), app(succ, 0))), c_app(ack, c_app(succ, x11))), x), xs) [2]
   app(c_app(filter, c_app(ack, 0)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(succ, x)), c_app(ack, 0)), x), xs) [2]
   app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, c_app(f, x)), f), x), xs) [1]
   app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, null_app), f), x), xs) [1]
   app(c_app(ack, c_app(succ, x)), c_app(succ, y)) -> app(c_app(ack, x), app(app(ack, c_app(succ, x)), y)) [1]
   app(c_app(ack, c_app(succ, x)), c_app(succ, y)) -> app(c_app(ack, x), app(app(ack, null_app), y)) [1]
   app(c_app(ack, c_app(succ, x)), c_app(succ, y)) -> app(null_app, app(app(ack, c_app(succ, x)), y)) [1]
   app(c_app(ack, c_app(succ, x)), c_app(succ, y)) -> app(null_app, app(app(ack, null_app), y)) [1]
   app(c_app(ack, c_app(succ, x)), y) -> app(c_app(ack, x), c_app(succ, 0)) [1]
   app(c_app(ack, c_app(succ, x)), y) -> app(c_app(ack, x), null_app) [1]
   app(c_app(ack, c_app(succ, x)), y) -> app(null_app, c_app(succ, 0)) [1]
   app(c_app(ack, c_app(succ, x)), y) -> app(null_app, null_app) [1]
   app(c_app(ack, 0), y) -> app(succ, y) [1]
   encArg(ack) -> ack [0]
   encArg(0) -> 0 [0]
   encArg(succ) -> succ [0]
   encArg(map) -> map [0]
   encArg(nil) -> nil [0]
   encArg(cons) -> cons [0]
   encArg(filter) -> filter [0]
   encArg(filter2) -> filter2 [0]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0]
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0]
   encode_ack -> ack [0]
   encode_0 -> 0 [0]
   encode_succ -> succ [0]
   encode_map -> map [0]
   encode_nil -> nil [0]
   encode_cons -> cons [0]
   encode_filter -> filter [0]
   encode_filter2 -> filter2 [0]
   encode_true -> true [0]
   encode_false -> false [0]
   app(x0, x1) -> c_app(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_app(v0, v1) -> null_encode_app [0]
   encode_ack -> null_encode_ack [0]
   encode_0 -> null_encode_0 [0]
   encode_succ -> null_encode_succ [0]
   encode_map -> null_encode_map [0]
   encode_nil -> null_encode_nil [0]
   encode_cons -> null_encode_cons [0]
   encode_filter -> null_encode_filter [0]
   encode_filter2 -> null_encode_filter2 [0]
   encode_true -> null_encode_true [0]
   encode_false -> null_encode_false [0]
   app(v0, v1) -> null_app [0]

The TRS has the following type information:
app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
c_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encArg :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
cons_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encArg :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app

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

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

   map => 6
   cons => 2
   filter2 => 5
   true => 9
   filter => 4
   false => 3
   nil => 7
   ack => 1
   succ => 8
   0 => 0
   null_encArg => 0
   null_encode_app => 0
   null_encode_ack => 0
   null_encode_0 => 0
   null_encode_succ => 0
   null_encode_map => 0
   null_encode_nil => 0
   null_encode_cons => 0
   null_encode_filter => 0
   null_encode_filter2 => 0
   null_encode_true => 0
   null_encode_false => 0
   null_app => 0

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

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

   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(app(app(5, app(f10, x9)), f10), x9), xs2)), 1 + 4 + f10), 1 + (1 + 2 + x9) + xs2), xs) :|: xs >= 0, xs2 >= 0, f10 >= 0, z = 1 + 4 + (1 + 4 + f10), z' = 1 + (1 + 2 + (1 + (1 + 2 + x9) + xs2)) + xs, x9 >= 0
   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(4, f7), x)), 1 + (1 + (1 + 5 + 3) + f7) + x8), x), xs) :|: xs >= 0, x8 >= 0, x >= 0, f7 >= 0, z = 1 + 4 + (1 + (1 + (1 + 5 + 3) + f7) + x8), z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(2, x7), app(app(4, f6), x))), 1 + (1 + (1 + 5 + 9) + f6) + x7), x), xs) :|: xs >= 0, x7 >= 0, x >= 0, f6 >= 0, z = 1 + 4 + (1 + (1 + (1 + 5 + 9) + f6) + x7), z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(2, app(f5, x6)), app(app(6, f5), xs1))), 1 + 6 + f5), 1 + (1 + 2 + x6) + xs1), xs) :|: xs >= 0, z' = 1 + (1 + 2 + (1 + (1 + 2 + x6) + xs1)) + xs, x6 >= 0, xs1 >= 0, f5 >= 0, z = 1 + 4 + (1 + 6 + f5)
   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(1, x10), app(app(1, app(8, x10)), y''))), 1 + 1 + (1 + 8 + x10)), 1 + 8 + y''), xs) :|: xs >= 0, z' = 1 + (1 + 2 + (1 + 8 + y'')) + xs, z = 1 + 4 + (1 + 1 + (1 + 8 + x10)), y'' >= 0, x10 >= 0
   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(1, x11), app(8, 0))), 1 + 1 + (1 + 8 + x11)), x), xs) :|: xs >= 0, z = 1 + 4 + (1 + 1 + (1 + 8 + x11)), x >= 0, x11 >= 0, z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 2 }-> app(app(app(app(5, app(8, x)), 1 + 1 + 0), x), xs) :|: xs >= 0, x >= 0, z = 1 + 4 + (1 + 1 + 0), z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 2 }-> app(app(app(app(5, 7), 1 + 6 + f8), 7), xs) :|: f8 >= 0, xs >= 0, z = 1 + 4 + (1 + 6 + f8), z' = 1 + (1 + 2 + 7) + xs
   app(z, z') -{ 2 }-> app(app(app(app(5, 7), 1 + 4 + f9), 7), xs) :|: xs >= 0, z' = 1 + (1 + 2 + 7) + xs, z = 1 + 4 + (1 + 4 + f9), f9 >= 0
   app(z, z') -{ 1 }-> app(app(app(app(5, 0), f), x), xs) :|: f >= 0, xs >= 0, x >= 0, z = 1 + 4 + f, z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 1 }-> app(app(app(app(5, 1 + f + x), f), x), xs) :|: f >= 0, xs >= 0, x >= 0, z = 1 + 4 + f, z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 1 }-> app(app(2, app(f, x)), app(app(6, f), xs)) :|: f >= 0, xs >= 0, x >= 0, z = 1 + 6 + f, z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 1 }-> app(8, y) :|: z = 1 + 1 + 0, y >= 0, z' = y
   app(z, z') -{ 1 }-> app(0, xs) :|: f >= 0, xs >= 0, z = 1 + (1 + (1 + 5 + 3) + f) + x, x >= 0, z' = xs
   app(z, z') -{ 1 }-> app(0, app(app(1, 0), y)) :|: z = 1 + 1 + (1 + 8 + x), x >= 0, y >= 0, z' = 1 + 8 + y
   app(z, z') -{ 1 }-> app(0, app(app(1, 1 + 8 + x), y)) :|: z = 1 + 1 + (1 + 8 + x), x >= 0, y >= 0, z' = 1 + 8 + y
   app(z, z') -{ 1 }-> app(0, app(0, xs)) :|: f >= 0, xs >= 0, x >= 0, z' = xs, z = 1 + (1 + (1 + 5 + 9) + f) + x
   app(z, z') -{ 1 }-> app(0, app(1 + 4 + f, xs)) :|: f >= 0, xs >= 0, x >= 0, z' = xs, z = 1 + (1 + (1 + 5 + 9) + f) + x
   app(z, z') -{ 1 }-> app(0, 0) :|: z = 1 + 1 + (1 + 8 + x), x >= 0, y >= 0, z' = y
   app(z, z') -{ 1 }-> app(0, 1 + 8 + 0) :|: z = 1 + 1 + (1 + 8 + x), x >= 0, y >= 0, z' = y
   app(z, z') -{ 1 }-> app(1 + 4 + f, xs) :|: f >= 0, xs >= 0, z = 1 + (1 + (1 + 5 + 3) + f) + x, x >= 0, z' = xs
   app(z, z') -{ 1 }-> app(1 + 2 + x, app(0, xs)) :|: f >= 0, xs >= 0, x >= 0, z' = xs, z = 1 + (1 + (1 + 5 + 9) + f) + x
   app(z, z') -{ 1 }-> app(1 + 2 + x, app(1 + 4 + f, xs)) :|: f >= 0, xs >= 0, x >= 0, z' = xs, z = 1 + (1 + (1 + 5 + 9) + f) + x
   app(z, z') -{ 1 }-> app(1 + 1 + x, app(app(1, 0), y)) :|: z = 1 + 1 + (1 + 8 + x), x >= 0, y >= 0, z' = 1 + 8 + y
   app(z, z') -{ 1 }-> app(1 + 1 + x, app(app(1, 1 + 8 + x), y)) :|: z = 1 + 1 + (1 + 8 + x), x >= 0, y >= 0, z' = 1 + 8 + y
   app(z, z') -{ 1 }-> app(1 + 1 + x, 0) :|: z = 1 + 1 + (1 + 8 + x), x >= 0, y >= 0, z' = y
   app(z, z') -{ 1 }-> app(1 + 1 + x, 1 + 8 + 0) :|: z = 1 + 1 + (1 + 8 + x), x >= 0, y >= 0, z' = y
   app(z, z') -{ 1 }-> 7 :|: f >= 0, z = 1 + 6 + f, z' = 7
   app(z, z') -{ 1 }-> 7 :|: f >= 0, z = 1 + 4 + f, z' = 7
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1
   encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 9 :|: z = 9
   encArg(z) -{ 0 }-> 8 :|: z = 8
   encArg(z) -{ 0 }-> 7 :|: z = 7
   encArg(z) -{ 0 }-> 6 :|: z = 6
   encArg(z) -{ 0 }-> 5 :|: z = 5
   encArg(z) -{ 0 }-> 4 :|: z = 4
   encArg(z) -{ 0 }-> 3 :|: z = 3
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_0 -{ 0 }-> 0 :|: 
   encode_ack -{ 0 }-> 1 :|: 
   encode_ack -{ 0 }-> 0 :|: 
   encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_cons -{ 0 }-> 2 :|: 
   encode_cons -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 3 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_filter -{ 0 }-> 4 :|: 
   encode_filter -{ 0 }-> 0 :|: 
   encode_filter2 -{ 0 }-> 5 :|: 
   encode_filter2 -{ 0 }-> 0 :|: 
   encode_map -{ 0 }-> 6 :|: 
   encode_map -{ 0 }-> 0 :|: 
   encode_nil -{ 0 }-> 7 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_succ -{ 0 }-> 8 :|: 
   encode_succ -{ 0 }-> 0 :|: 
   encode_true -{ 0 }-> 9 :|: 
   encode_true -{ 0 }-> 0 :|: 


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

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

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

   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(app(app(5, app(z - 10, x9)), z - 10), x9), xs2)), 1 + 4 + (z - 10)), 1 + (1 + 2 + x9) + xs2), xs) :|: xs >= 0, xs2 >= 0, z - 10 >= 0, z' = 1 + (1 + 2 + (1 + (1 + 2 + x9) + xs2)) + xs, x9 >= 0
   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(4, f7), x)), 1 + (1 + (1 + 5 + 3) + f7) + x8), x), xs) :|: xs >= 0, x8 >= 0, x >= 0, f7 >= 0, z = 1 + 4 + (1 + (1 + (1 + 5 + 3) + f7) + x8), z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(2, x7), app(app(4, f6), x))), 1 + (1 + (1 + 5 + 9) + f6) + x7), x), xs) :|: xs >= 0, x7 >= 0, x >= 0, f6 >= 0, z = 1 + 4 + (1 + (1 + (1 + 5 + 9) + f6) + x7), z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(2, app(z - 12, x6)), app(app(6, z - 12), xs1))), 1 + 6 + (z - 12)), 1 + (1 + 2 + x6) + xs1), xs) :|: xs >= 0, z' = 1 + (1 + 2 + (1 + (1 + 2 + x6) + xs1)) + xs, x6 >= 0, xs1 >= 0, z - 12 >= 0
   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(1, z - 16), app(app(1, app(8, z - 16)), y''))), 1 + 1 + (1 + 8 + (z - 16))), 1 + 8 + y''), xs) :|: xs >= 0, z' = 1 + (1 + 2 + (1 + 8 + y'')) + xs, y'' >= 0, z - 16 >= 0
   app(z, z') -{ 2 }-> app(app(app(app(5, app(app(1, z - 16), app(8, 0))), 1 + 1 + (1 + 8 + (z - 16))), x), xs) :|: xs >= 0, x >= 0, z - 16 >= 0, z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 2 }-> app(app(app(app(5, app(8, x)), 1 + 1 + 0), x), xs) :|: xs >= 0, x >= 0, z = 1 + 4 + (1 + 1 + 0), z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 2 }-> app(app(app(app(5, 7), 1 + 6 + (z - 12)), 7), z' - 11) :|: z - 12 >= 0, z' - 11 >= 0
   app(z, z') -{ 2 }-> app(app(app(app(5, 7), 1 + 4 + (z - 10)), 7), z' - 11) :|: z' - 11 >= 0, z - 10 >= 0
   app(z, z') -{ 1 }-> app(app(app(app(5, 0), z - 5), x), xs) :|: z - 5 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 1 }-> app(app(app(app(5, 1 + (z - 5) + x), z - 5), x), xs) :|: z - 5 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 1 }-> app(app(2, app(z - 7, x)), app(app(6, z - 7), xs)) :|: z - 7 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 1 }-> app(8, z') :|: z = 1 + 1 + 0, z' >= 0
   app(z, z') -{ 1 }-> app(0, z') :|: f >= 0, z' >= 0, z = 1 + (1 + (1 + 5 + 3) + f) + x, x >= 0
   app(z, z') -{ 1 }-> app(0, app(app(1, 0), z' - 9)) :|: z - 11 >= 0, z' - 9 >= 0
   app(z, z') -{ 1 }-> app(0, app(app(1, 1 + 8 + (z - 11)), z' - 9)) :|: z - 11 >= 0, z' - 9 >= 0
   app(z, z') -{ 1 }-> app(0, app(0, z')) :|: f >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 9) + f) + x
   app(z, z') -{ 1 }-> app(0, app(1 + 4 + f, z')) :|: f >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 9) + f) + x
   app(z, z') -{ 1 }-> app(0, 0) :|: z - 11 >= 0, z' >= 0
   app(z, z') -{ 1 }-> app(0, 1 + 8 + 0) :|: z - 11 >= 0, z' >= 0
   app(z, z') -{ 1 }-> app(1 + 4 + f, z') :|: f >= 0, z' >= 0, z = 1 + (1 + (1 + 5 + 3) + f) + x, x >= 0
   app(z, z') -{ 1 }-> app(1 + 2 + x, app(0, z')) :|: f >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 9) + f) + x
   app(z, z') -{ 1 }-> app(1 + 2 + x, app(1 + 4 + f, z')) :|: f >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 9) + f) + x
   app(z, z') -{ 1 }-> app(1 + 1 + (z - 11), app(app(1, 0), z' - 9)) :|: z - 11 >= 0, z' - 9 >= 0
   app(z, z') -{ 1 }-> app(1 + 1 + (z - 11), app(app(1, 1 + 8 + (z - 11)), z' - 9)) :|: z - 11 >= 0, z' - 9 >= 0
   app(z, z') -{ 1 }-> app(1 + 1 + (z - 11), 0) :|: z - 11 >= 0, z' >= 0
   app(z, z') -{ 1 }-> app(1 + 1 + (z - 11), 1 + 8 + 0) :|: z - 11 >= 0, z' >= 0
   app(z, z') -{ 1 }-> 7 :|: z - 7 >= 0, z' = 7
   app(z, z') -{ 1 }-> 7 :|: z - 5 >= 0, z' = 7
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 9 :|: z = 9
   encArg(z) -{ 0 }-> 8 :|: z = 8
   encArg(z) -{ 0 }-> 7 :|: z = 7
   encArg(z) -{ 0 }-> 6 :|: z = 6
   encArg(z) -{ 0 }-> 5 :|: z = 5
   encArg(z) -{ 0 }-> 4 :|: z = 4
   encArg(z) -{ 0 }-> 3 :|: z = 3
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_ack -{ 0 }-> 1 :|: 
   encode_ack -{ 0 }-> 0 :|: 
   encode_app(z, z') -{ 0 }-> app(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons -{ 0 }-> 2 :|: 
   encode_cons -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 3 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_filter -{ 0 }-> 4 :|: 
   encode_filter -{ 0 }-> 0 :|: 
   encode_filter2 -{ 0 }-> 5 :|: 
   encode_filter2 -{ 0 }-> 0 :|: 
   encode_map -{ 0 }-> 6 :|: 
   encode_map -{ 0 }-> 0 :|: 
   encode_nil -{ 0 }-> 7 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_succ -{ 0 }-> 8 :|: 
   encode_succ -{ 0 }-> 0 :|: 
   encode_true -{ 0 }-> 9 :|: 
   encode_true -{ 0 }-> 0 :|: 


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

(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:

   encArg(v0) -> null_encArg [0]
   encode_app(v0, v1) -> null_encode_app [0]
   encode_ack -> null_encode_ack [0]
   encode_0 -> null_encode_0 [0]
   encode_succ -> null_encode_succ [0]
   encode_map -> null_encode_map [0]
   encode_nil -> null_encode_nil [0]
   encode_cons -> null_encode_cons [0]
   encode_filter -> null_encode_filter [0]
   encode_filter2 -> null_encode_filter2 [0]
   encode_true -> null_encode_true [0]
   encode_false -> null_encode_false [0]
   app(v0, v1) -> null_app [0]

And the following fresh constants:    null_encArg, null_encode_app, null_encode_ack, null_encode_0, null_encode_succ, null_encode_map, null_encode_nil, null_encode_cons, null_encode_filter, null_encode_filter2, null_encode_true, null_encode_false, null_app

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

(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:

   app(c_app(map, f), c_app(c_app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, true), f), x), xs) -> app(app(cons, x), app(app(filter, f), xs)) [1]
   app(c_app(c_app(c_app(filter2, false), f), x), xs) -> app(app(filter, f), xs) [1]
   app(c_app(map, f), nil) -> nil [1]
   app(c_app(filter, f), nil) -> nil [1]
   app(c_app(filter, f), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(f, x)), f), x), xs) [1]
   app(c_app(ack, c_app(succ, x)), c_app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y)) [1]
   app(c_app(ack, c_app(succ, x)), y) -> app(app(ack, x), app(succ, 0)) [1]
   app(c_app(ack, 0), y) -> app(succ, y) [1]
   encArg(ack) -> ack [0]
   encArg(0) -> 0 [0]
   encArg(succ) -> succ [0]
   encArg(map) -> map [0]
   encArg(nil) -> nil [0]
   encArg(cons) -> cons [0]
   encArg(filter) -> filter [0]
   encArg(filter2) -> filter2 [0]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0]
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0]
   encode_ack -> ack [0]
   encode_0 -> 0 [0]
   encode_succ -> succ [0]
   encode_map -> map [0]
   encode_nil -> nil [0]
   encode_cons -> cons [0]
   encode_filter -> filter [0]
   encode_filter2 -> filter2 [0]
   encode_true -> true [0]
   encode_false -> false [0]
   app(x0, x1) -> c_app(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_app(v0, v1) -> null_encode_app [0]
   encode_ack -> null_encode_ack [0]
   encode_0 -> null_encode_0 [0]
   encode_succ -> null_encode_succ [0]
   encode_map -> null_encode_map [0]
   encode_nil -> null_encode_nil [0]
   encode_cons -> null_encode_cons [0]
   encode_filter -> null_encode_filter [0]
   encode_filter2 -> null_encode_filter2 [0]
   encode_true -> null_encode_true [0]
   encode_false -> null_encode_false [0]
   app(v0, v1) -> null_app [0]

The TRS has the following type information:
app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
c_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encArg :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
cons_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
encode_false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encArg :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_ack :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_succ :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_map :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_true :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_encode_false :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app
null_app :: map:c_app:cons:filter2:true:filter:false:nil:ack:succ:0:cons_app:null_encArg:null_encode_app:null_encode_ack:null_encode_0:null_encode_succ:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app

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:

   map => 6
   cons => 2
   filter2 => 5
   true => 9
   filter => 4
   false => 3
   nil => 7
   ack => 1
   succ => 8
   0 => 0
   null_encArg => 0
   null_encode_app => 0
   null_encode_ack => 0
   null_encode_0 => 0
   null_encode_succ => 0
   null_encode_map => 0
   null_encode_nil => 0
   null_encode_cons => 0
   null_encode_filter => 0
   null_encode_filter2 => 0
   null_encode_true => 0
   null_encode_false => 0
   null_app => 0

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

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

   app(z, z') -{ 1 }-> app(app(app(app(5, app(f, x)), f), x), xs) :|: f >= 0, xs >= 0, x >= 0, z = 1 + 4 + f, z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 1 }-> app(app(4, f), xs) :|: f >= 0, xs >= 0, z = 1 + (1 + (1 + 5 + 3) + f) + x, x >= 0, z' = xs
   app(z, z') -{ 1 }-> app(app(2, x), app(app(4, f), xs)) :|: f >= 0, xs >= 0, x >= 0, z' = xs, z = 1 + (1 + (1 + 5 + 9) + f) + x
   app(z, z') -{ 1 }-> app(app(2, app(f, x)), app(app(6, f), xs)) :|: f >= 0, xs >= 0, x >= 0, z = 1 + 6 + f, z' = 1 + (1 + 2 + x) + xs
   app(z, z') -{ 1 }-> app(app(1, x), app(app(1, app(8, x)), y)) :|: z = 1 + 1 + (1 + 8 + x), x >= 0, y >= 0, z' = 1 + 8 + y
   app(z, z') -{ 1 }-> app(app(1, x), app(8, 0)) :|: z = 1 + 1 + (1 + 8 + x), x >= 0, y >= 0, z' = y
   app(z, z') -{ 1 }-> app(8, y) :|: z = 1 + 1 + 0, y >= 0, z' = y
   app(z, z') -{ 1 }-> 7 :|: f >= 0, z = 1 + 6 + f, z' = 7
   app(z, z') -{ 1 }-> 7 :|: f >= 0, z = 1 + 4 + f, z' = 7
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1
   encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 9 :|: z = 9
   encArg(z) -{ 0 }-> 8 :|: z = 8
   encArg(z) -{ 0 }-> 7 :|: z = 7
   encArg(z) -{ 0 }-> 6 :|: z = 6
   encArg(z) -{ 0 }-> 5 :|: z = 5
   encArg(z) -{ 0 }-> 4 :|: z = 4
   encArg(z) -{ 0 }-> 3 :|: z = 3
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_0 -{ 0 }-> 0 :|: 
   encode_ack -{ 0 }-> 1 :|: 
   encode_ack -{ 0 }-> 0 :|: 
   encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_cons -{ 0 }-> 2 :|: 
   encode_cons -{ 0 }-> 0 :|: 
   encode_false -{ 0 }-> 3 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_filter -{ 0 }-> 4 :|: 
   encode_filter -{ 0 }-> 0 :|: 
   encode_filter2 -{ 0 }-> 5 :|: 
   encode_filter2 -{ 0 }-> 0 :|: 
   encode_map -{ 0 }-> 6 :|: 
   encode_map -{ 0 }-> 0 :|: 
   encode_nil -{ 0 }-> 7 :|: 
   encode_nil -{ 0 }-> 0 :|: 
   encode_succ -{ 0 }-> 8 :|: 
   encode_succ -{ 0 }-> 0 :|: 
   encode_true -{ 0 }-> 9 :|: 
   encode_true -{ 0 }-> 0 :|: 

Only complete derivations are relevant for the runtime complexity.

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

(35) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   ENCARG(ack) -> c
   ENCARG(0) -> c1
   ENCARG(succ) -> c2
   ENCARG(map) -> c3
   ENCARG(nil) -> c4
   ENCARG(cons) -> c5
   ENCARG(filter) -> c6
   ENCARG(filter2) -> c7
   ENCARG(true) -> c8
   ENCARG(false) -> c9
   ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_ACK -> c12
   ENCODE_0 -> c13
   ENCODE_SUCC -> c14
   ENCODE_MAP -> c15
   ENCODE_NIL -> c16
   ENCODE_CONS -> c17
   ENCODE_FILTER -> c18
   ENCODE_FILTER2 -> c19
   ENCODE_TRUE -> c20
   ENCODE_FALSE -> c21
   APP(app(ack, 0), z0) -> c22(APP(succ, z0))
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)), APP(ack, z0), APP(succ, 0))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(ack, z0), APP(app(ack, app(succ, z0)), z1), APP(ack, app(succ, z0)), APP(succ, z0))
   APP(app(map, z0), nil) -> c25
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0))
   APP(app(filter, z0), nil) -> c27
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2), APP(filter, z0))
S tuples:
   APP(app(ack, 0), z0) -> c22(APP(succ, z0))
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)), APP(ack, z0), APP(succ, 0))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(ack, z0), APP(app(ack, app(succ, z0)), z1), APP(ack, app(succ, z0)), APP(succ, z0))
   APP(app(map, z0), nil) -> c25
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0))
   APP(app(filter, z0), nil) -> c27
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2), APP(filter, z0))
K tuples:none
Defined Rule Symbols:   app_2, encArg_1, encode_app_2, encode_ack, encode_0, encode_succ, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false

Defined Pair Symbols:   ENCARG_1, ENCODE_APP_2, ENCODE_ACK, ENCODE_0, ENCODE_SUCC, ENCODE_MAP, ENCODE_NIL, ENCODE_CONS, ENCODE_FILTER, ENCODE_FILTER2, ENCODE_TRUE, ENCODE_FALSE, APP_2

Compound Symbols:   c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10_3, c11_3, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22_1, c23_3, c24_5, c25, c26_5, c27, c28_5, c29_4, c30_2


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

(37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 20 trailing nodes:
   ENCODE_FILTER2 -> c19
   ENCODE_MAP -> c15
   ENCODE_ACK -> c12
   ENCODE_TRUE -> c20
   ENCODE_0 -> c13
   ENCARG(true) -> c8
   ENCARG(0) -> c1
   ENCODE_NIL -> c16
   ENCARG(ack) -> c
   ENCARG(filter) -> c6
   ENCARG(nil) -> c4
   ENCARG(filter2) -> c7
   ENCODE_SUCC -> c14
   ENCARG(succ) -> c2
   ENCARG(cons) -> c5
   ENCARG(map) -> c3
   ENCODE_FILTER -> c18
   ENCODE_CONS -> c17
   ENCARG(false) -> c9
   ENCODE_FALSE -> c21

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(app(ack, 0), z0) -> c22(APP(succ, z0))
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)), APP(ack, z0), APP(succ, 0))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(ack, z0), APP(app(ack, app(succ, z0)), z1), APP(ack, app(succ, z0)), APP(succ, z0))
   APP(app(map, z0), nil) -> c25
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0))
   APP(app(filter, z0), nil) -> c27
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2), APP(filter, z0))
S tuples:
   APP(app(ack, 0), z0) -> c22(APP(succ, z0))
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)), APP(ack, z0), APP(succ, 0))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(ack, z0), APP(app(ack, app(succ, z0)), z1), APP(ack, app(succ, z0)), APP(succ, z0))
   APP(app(map, z0), nil) -> c25
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(app(cons, app(z0, z1)), app(app(map, z0), z2)), APP(cons, app(z0, z1)), APP(z0, z1), APP(app(map, z0), z2), APP(map, z0))
   APP(app(filter, z0), nil) -> c27
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(app(app(filter2, app(z0, z1)), z0), z1), APP(app(filter2, app(z0, z1)), z0), APP(filter2, app(z0, z1)), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2), APP(filter, z0))
K tuples:none
Defined Rule Symbols:   app_2, encArg_1, encode_app_2, encode_ack, encode_0, encode_succ, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false

Defined Pair Symbols:   ENCARG_1, ENCODE_APP_2, APP_2

Compound Symbols:   c10_3, c11_3, c22_1, c23_3, c24_5, c25, c26_5, c27, c28_5, c29_4, c30_2


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

(39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 16 trailing tuple parts
----------------------------------------

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
K tuples:none
Defined Rule Symbols:   app_2, encArg_1, encode_app_2, encode_ack, encode_0, encode_succ, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false

Defined Pair Symbols:   ENCARG_1, ENCODE_APP_2, APP_2

Compound Symbols:   c10_3, c11_3, c25, c27, c22, c23_1, c24_2, c26_2, c28_2, c29_1, c30_1


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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
   ENCODE_APP(z0, z1) -> c(ENCARG(z0))
   ENCODE_APP(z0, z1) -> c(ENCARG(z1))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
K tuples:none
Defined Rule Symbols:   app_2, encArg_1, encode_app_2, encode_ack, encode_0, encode_succ, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false

Defined Pair Symbols:   ENCARG_1, APP_2, ENCODE_APP_2

Compound Symbols:   c10_3, c25, c27, c22, c23_1, c24_2, c26_2, c28_2, c29_1, c30_1, c_1


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

(43) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_APP(z0, z1) -> c(ENCARG(z0))
   ENCODE_APP(z0, z1) -> c(ENCARG(z1))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
K tuples:none
Defined Rule Symbols:   app_2, encArg_1, encode_app_2, encode_ack, encode_0, encode_succ, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false

Defined Pair Symbols:   ENCARG_1, APP_2, ENCODE_APP_2

Compound Symbols:   c10_3, c25, c27, c22, c23_1, c24_2, c26_2, c28_2, c29_1, c30_1, c_1


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

(45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_ack -> ack
   encode_0 -> 0
   encode_succ -> succ
   encode_map -> map
   encode_nil -> nil
   encode_cons -> cons
   encode_filter -> filter
   encode_filter2 -> filter2
   encode_true -> true
   encode_false -> false

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   ENCARG_1, APP_2, ENCODE_APP_2

Compound Symbols:   c10_3, c25, c27, c22, c23_1, c24_2, c26_2, c28_2, c29_1, c30_1, c_1


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(z0, z1)) -> c10(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by 
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0), ENCARG(ack))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0), ENCARG(succ))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0), ENCARG(map))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0), ENCARG(true))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0), ENCARG(false))
   ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1)))
   ENCARG(cons_app(ack, x1)) -> c10(APP(ack, encArg(x1)), ENCARG(ack), ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(APP(succ, encArg(x1)), ENCARG(succ), ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1))
   ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0), ENCARG(ack))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0), ENCARG(succ))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0), ENCARG(map))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0), ENCARG(true))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0), ENCARG(false))
   ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1)))
   ENCARG(cons_app(ack, x1)) -> c10(APP(ack, encArg(x1)), ENCARG(ack), ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(APP(succ, encArg(x1)), ENCARG(succ), ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1))
   ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCODE_APP_2, ENCARG_1

Compound Symbols:   c25, c27, c22, c23_1, c24_2, c26_2, c28_2, c29_1, c30_1, c_1, c10_3


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

(49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 30 trailing tuple parts
----------------------------------------

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
   ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1)))
   ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCODE_APP_2, ENCARG_1

Compound Symbols:   c25, c27, c22, c23_1, c24_2, c26_2, c28_2, c29_1, c30_1, c_1, c10_3, c10_2, c10_1


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, app(succ, z0)), z1)), APP(app(ack, app(succ, z0)), z1)) by 
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
   ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1)))
   ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCODE_APP_2, ENCARG_1

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c28_2, c29_1, c30_1, c_1, c10_3, c10_2, c10_1, c24_2


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APP(app(filter, z0), app(app(cons, z1), z2)) -> c28(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) by 
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(app(app(filter2, app(succ, z0)), app(ack, 0)), z0), x2), APP(app(ack, 0), z0))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(app(app(filter2, nil), app(map, z0)), nil), x2), APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2), APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(app(app(filter2, nil), app(filter, z0)), nil), x2), APP(app(filter, z0), nil))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2), APP(app(app(app(filter2, true), z0), z1), z2))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
   ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1)))
   ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(app(app(filter2, app(succ, z0)), app(ack, 0)), z0), x2), APP(app(ack, 0), z0))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(app(app(filter2, nil), app(map, z0)), nil), x2), APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2), APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(app(app(filter2, nil), app(filter, z0)), nil), x2), APP(app(filter, z0), nil))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2), APP(app(app(app(filter2, true), z0), z1), z2))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(app(app(filter2, app(succ, z0)), app(ack, 0)), z0), x2), APP(app(ack, 0), z0))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(app(app(filter2, nil), app(map, z0)), nil), x2), APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(cons, app(z0, z1)), app(app(map, z0), z2))), app(map, z0)), app(app(cons, z1), z2)), x2), APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(app(app(filter2, nil), app(filter, z0)), nil), x2), APP(app(filter, z0), nil))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(cons, z1), app(app(filter, z0), z2))), app(app(app(filter2, true), z0), z1)), z2), x2), APP(app(app(app(filter2, true), z0), z1), z2))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCODE_APP_2, ENCARG_1

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c_1, c10_3, c10_2, c10_1, c24_2, c28_2


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

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
   ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1)))
   ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCODE_APP_2, ENCARG_1

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c_1, c10_3, c10_2, c10_1, c24_2, c28_2, c28_1


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) by 
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(ack, x1) -> c(APP(ack, encArg(x1)))
   ENCODE_APP(0, x1) -> c(APP(0, encArg(x1)))
   ENCODE_APP(succ, x1) -> c(APP(succ, encArg(x1)))
   ENCODE_APP(map, x1) -> c(APP(map, encArg(x1)))
   ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1)))
   ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1)))
   ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1)))
   ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1)))
   ENCODE_APP(true, x1) -> c(APP(true, encArg(x1)))
   ENCODE_APP(false, x1) -> c(APP(false, encArg(x1)))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1)))
   ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(ack, x1) -> c(APP(ack, encArg(x1)))
   ENCODE_APP(0, x1) -> c(APP(0, encArg(x1)))
   ENCODE_APP(succ, x1) -> c(APP(succ, encArg(x1)))
   ENCODE_APP(map, x1) -> c(APP(map, encArg(x1)))
   ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1)))
   ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1)))
   ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1)))
   ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1)))
   ENCODE_APP(true, x1) -> c(APP(true, encArg(x1)))
   ENCODE_APP(false, x1) -> c(APP(false, encArg(x1)))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_3, c10_2, c10_1, c24_2, c28_2, c28_1, c_1


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

(59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCODE_APP(ack, x1) -> c(APP(ack, encArg(x1)))
   ENCODE_APP(false, x1) -> c(APP(false, encArg(x1)))
   ENCODE_APP(0, x1) -> c(APP(0, encArg(x1)))
   ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1)))
   ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1)))
   ENCODE_APP(succ, x1) -> c(APP(succ, encArg(x1)))
   ENCODE_APP(map, x1) -> c(APP(map, encArg(x1)))
   ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1)))
   ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1)))
   ENCODE_APP(true, x1) -> c(APP(true, encArg(x1)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1)))
   ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_3, c10_2, c10_1, c24_2, c28_2, c28_1, c_1


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, cons_app(z0, z1))) -> c10(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) by 
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(APP(ack, app(encArg(x1), encArg(x2))), ENCARG(ack), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(APP(0, app(encArg(x1), encArg(x2))), ENCARG(0), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(APP(succ, app(encArg(x1), encArg(x2))), ENCARG(succ), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(APP(ack, app(encArg(x1), encArg(x2))), ENCARG(ack), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(APP(0, app(encArg(x1), encArg(x2))), ENCARG(0), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(APP(succ, app(encArg(x1), encArg(x2))), ENCARG(succ), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_3, c10_2, c10_1, c24_2, c28_2, c28_1, c_1


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

(63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 20 trailing tuple parts
----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_3, c10_2, c10_1, c24_2, c28_2, c28_1, c_1


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(cons_app(z0, z1), x1)) -> c10(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) by 
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)), ENCARG(ack))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)), ENCARG(0))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)), ENCARG(succ))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false))
   ENCARG(cons_app(cons_app(x0, x1), cons_app(z0, z1))) -> c10(APP(app(encArg(x0), encArg(x1)), app(encArg(z0), encArg(z1))), ENCARG(cons_app(x0, x1)), ENCARG(cons_app(z0, z1)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(APP(app(0, encArg(x1)), encArg(x2)), ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(APP(app(succ, encArg(x1)), encArg(x2)), ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)), ENCARG(ack))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)), ENCARG(0))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)), ENCARG(succ))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(APP(app(0, encArg(x1)), encArg(x2)), ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(APP(app(succ, encArg(x1)), encArg(x2)), ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 17 trailing tuple parts
----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, ack)) -> c10(APP(encArg(x0), ack), ENCARG(x0)) by 
   ENCARG(cons_app(ack, ack)) -> c10(APP(ack, ack), ENCARG(ack))
   ENCARG(cons_app(0, ack)) -> c10(APP(0, ack), ENCARG(0))
   ENCARG(cons_app(succ, ack)) -> c10(APP(succ, ack), ENCARG(succ))
   ENCARG(cons_app(map, ack)) -> c10(APP(map, ack), ENCARG(map))
   ENCARG(cons_app(nil, ack)) -> c10(APP(nil, ack), ENCARG(nil))
   ENCARG(cons_app(cons, ack)) -> c10(APP(cons, ack), ENCARG(cons))
   ENCARG(cons_app(filter, ack)) -> c10(APP(filter, ack), ENCARG(filter))
   ENCARG(cons_app(filter2, ack)) -> c10(APP(filter2, ack), ENCARG(filter2))
   ENCARG(cons_app(true, ack)) -> c10(APP(true, ack), ENCARG(true))
   ENCARG(cons_app(false, ack)) -> c10(APP(false, ack), ENCARG(false))
   ENCARG(cons_app(cons_app(z0, z1), ack)) -> c10(APP(app(encArg(z0), encArg(z1)), ack), ENCARG(cons_app(z0, z1)))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(ack, ack)) -> c10(APP(ack, ack), ENCARG(ack))
   ENCARG(cons_app(0, ack)) -> c10(APP(0, ack), ENCARG(0))
   ENCARG(cons_app(succ, ack)) -> c10(APP(succ, ack), ENCARG(succ))
   ENCARG(cons_app(map, ack)) -> c10(APP(map, ack), ENCARG(map))
   ENCARG(cons_app(nil, ack)) -> c10(APP(nil, ack), ENCARG(nil))
   ENCARG(cons_app(cons, ack)) -> c10(APP(cons, ack), ENCARG(cons))
   ENCARG(cons_app(filter, ack)) -> c10(APP(filter, ack), ENCARG(filter))
   ENCARG(cons_app(filter2, ack)) -> c10(APP(filter2, ack), ENCARG(filter2))
   ENCARG(cons_app(true, ack)) -> c10(APP(true, ack), ENCARG(true))
   ENCARG(cons_app(false, ack)) -> c10(APP(false, ack), ENCARG(false))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCARG(cons_app(nil, ack)) -> c10(APP(nil, ack), ENCARG(nil))
   ENCARG(cons_app(false, ack)) -> c10(APP(false, ack), ENCARG(false))
   ENCARG(cons_app(filter2, ack)) -> c10(APP(filter2, ack), ENCARG(filter2))
   ENCARG(cons_app(map, ack)) -> c10(APP(map, ack), ENCARG(map))
   ENCARG(cons_app(0, ack)) -> c10(APP(0, ack), ENCARG(0))
   ENCARG(cons_app(ack, ack)) -> c10(APP(ack, ack), ENCARG(ack))
   ENCARG(cons_app(cons, ack)) -> c10(APP(cons, ack), ENCARG(cons))
   ENCARG(cons_app(true, ack)) -> c10(APP(true, ack), ENCARG(true))
   ENCARG(cons_app(filter, ack)) -> c10(APP(filter, ack), ENCARG(filter))
   ENCARG(cons_app(succ, ack)) -> c10(APP(succ, ack), ENCARG(succ))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, 0)) -> c10(APP(encArg(x0), 0), ENCARG(x0)) by 
   ENCARG(cons_app(ack, 0)) -> c10(APP(ack, 0), ENCARG(ack))
   ENCARG(cons_app(0, 0)) -> c10(APP(0, 0), ENCARG(0))
   ENCARG(cons_app(succ, 0)) -> c10(APP(succ, 0), ENCARG(succ))
   ENCARG(cons_app(map, 0)) -> c10(APP(map, 0), ENCARG(map))
   ENCARG(cons_app(nil, 0)) -> c10(APP(nil, 0), ENCARG(nil))
   ENCARG(cons_app(cons, 0)) -> c10(APP(cons, 0), ENCARG(cons))
   ENCARG(cons_app(filter, 0)) -> c10(APP(filter, 0), ENCARG(filter))
   ENCARG(cons_app(filter2, 0)) -> c10(APP(filter2, 0), ENCARG(filter2))
   ENCARG(cons_app(true, 0)) -> c10(APP(true, 0), ENCARG(true))
   ENCARG(cons_app(false, 0)) -> c10(APP(false, 0), ENCARG(false))
   ENCARG(cons_app(cons_app(z0, z1), 0)) -> c10(APP(app(encArg(z0), encArg(z1)), 0), ENCARG(cons_app(z0, z1)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(ack, 0)) -> c10(APP(ack, 0), ENCARG(ack))
   ENCARG(cons_app(0, 0)) -> c10(APP(0, 0), ENCARG(0))
   ENCARG(cons_app(succ, 0)) -> c10(APP(succ, 0), ENCARG(succ))
   ENCARG(cons_app(map, 0)) -> c10(APP(map, 0), ENCARG(map))
   ENCARG(cons_app(nil, 0)) -> c10(APP(nil, 0), ENCARG(nil))
   ENCARG(cons_app(cons, 0)) -> c10(APP(cons, 0), ENCARG(cons))
   ENCARG(cons_app(filter, 0)) -> c10(APP(filter, 0), ENCARG(filter))
   ENCARG(cons_app(filter2, 0)) -> c10(APP(filter2, 0), ENCARG(filter2))
   ENCARG(cons_app(true, 0)) -> c10(APP(true, 0), ENCARG(true))
   ENCARG(cons_app(false, 0)) -> c10(APP(false, 0), ENCARG(false))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCARG(cons_app(filter, 0)) -> c10(APP(filter, 0), ENCARG(filter))
   ENCARG(cons_app(cons, 0)) -> c10(APP(cons, 0), ENCARG(cons))
   ENCARG(cons_app(0, 0)) -> c10(APP(0, 0), ENCARG(0))
   ENCARG(cons_app(filter2, 0)) -> c10(APP(filter2, 0), ENCARG(filter2))
   ENCARG(cons_app(map, 0)) -> c10(APP(map, 0), ENCARG(map))
   ENCARG(cons_app(ack, 0)) -> c10(APP(ack, 0), ENCARG(ack))
   ENCARG(cons_app(false, 0)) -> c10(APP(false, 0), ENCARG(false))
   ENCARG(cons_app(true, 0)) -> c10(APP(true, 0), ENCARG(true))
   ENCARG(cons_app(succ, 0)) -> c10(APP(succ, 0), ENCARG(succ))
   ENCARG(cons_app(nil, 0)) -> c10(APP(nil, 0), ENCARG(nil))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, succ)) -> c10(APP(encArg(x0), succ), ENCARG(x0)) by 
   ENCARG(cons_app(ack, succ)) -> c10(APP(ack, succ), ENCARG(ack))
   ENCARG(cons_app(0, succ)) -> c10(APP(0, succ), ENCARG(0))
   ENCARG(cons_app(succ, succ)) -> c10(APP(succ, succ), ENCARG(succ))
   ENCARG(cons_app(map, succ)) -> c10(APP(map, succ), ENCARG(map))
   ENCARG(cons_app(nil, succ)) -> c10(APP(nil, succ), ENCARG(nil))
   ENCARG(cons_app(cons, succ)) -> c10(APP(cons, succ), ENCARG(cons))
   ENCARG(cons_app(filter, succ)) -> c10(APP(filter, succ), ENCARG(filter))
   ENCARG(cons_app(filter2, succ)) -> c10(APP(filter2, succ), ENCARG(filter2))
   ENCARG(cons_app(true, succ)) -> c10(APP(true, succ), ENCARG(true))
   ENCARG(cons_app(false, succ)) -> c10(APP(false, succ), ENCARG(false))
   ENCARG(cons_app(cons_app(z0, z1), succ)) -> c10(APP(app(encArg(z0), encArg(z1)), succ), ENCARG(cons_app(z0, z1)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(ack, succ)) -> c10(APP(ack, succ), ENCARG(ack))
   ENCARG(cons_app(0, succ)) -> c10(APP(0, succ), ENCARG(0))
   ENCARG(cons_app(succ, succ)) -> c10(APP(succ, succ), ENCARG(succ))
   ENCARG(cons_app(map, succ)) -> c10(APP(map, succ), ENCARG(map))
   ENCARG(cons_app(nil, succ)) -> c10(APP(nil, succ), ENCARG(nil))
   ENCARG(cons_app(cons, succ)) -> c10(APP(cons, succ), ENCARG(cons))
   ENCARG(cons_app(filter, succ)) -> c10(APP(filter, succ), ENCARG(filter))
   ENCARG(cons_app(filter2, succ)) -> c10(APP(filter2, succ), ENCARG(filter2))
   ENCARG(cons_app(true, succ)) -> c10(APP(true, succ), ENCARG(true))
   ENCARG(cons_app(false, succ)) -> c10(APP(false, succ), ENCARG(false))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCARG(cons_app(cons, succ)) -> c10(APP(cons, succ), ENCARG(cons))
   ENCARG(cons_app(ack, succ)) -> c10(APP(ack, succ), ENCARG(ack))
   ENCARG(cons_app(false, succ)) -> c10(APP(false, succ), ENCARG(false))
   ENCARG(cons_app(filter, succ)) -> c10(APP(filter, succ), ENCARG(filter))
   ENCARG(cons_app(nil, succ)) -> c10(APP(nil, succ), ENCARG(nil))
   ENCARG(cons_app(true, succ)) -> c10(APP(true, succ), ENCARG(true))
   ENCARG(cons_app(0, succ)) -> c10(APP(0, succ), ENCARG(0))
   ENCARG(cons_app(succ, succ)) -> c10(APP(succ, succ), ENCARG(succ))
   ENCARG(cons_app(filter2, succ)) -> c10(APP(filter2, succ), ENCARG(filter2))
   ENCARG(cons_app(map, succ)) -> c10(APP(map, succ), ENCARG(map))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, map)) -> c10(APP(encArg(x0), map), ENCARG(x0)) by 
   ENCARG(cons_app(ack, map)) -> c10(APP(ack, map), ENCARG(ack))
   ENCARG(cons_app(0, map)) -> c10(APP(0, map), ENCARG(0))
   ENCARG(cons_app(succ, map)) -> c10(APP(succ, map), ENCARG(succ))
   ENCARG(cons_app(map, map)) -> c10(APP(map, map), ENCARG(map))
   ENCARG(cons_app(nil, map)) -> c10(APP(nil, map), ENCARG(nil))
   ENCARG(cons_app(cons, map)) -> c10(APP(cons, map), ENCARG(cons))
   ENCARG(cons_app(filter, map)) -> c10(APP(filter, map), ENCARG(filter))
   ENCARG(cons_app(filter2, map)) -> c10(APP(filter2, map), ENCARG(filter2))
   ENCARG(cons_app(true, map)) -> c10(APP(true, map), ENCARG(true))
   ENCARG(cons_app(false, map)) -> c10(APP(false, map), ENCARG(false))
   ENCARG(cons_app(cons_app(z0, z1), map)) -> c10(APP(app(encArg(z0), encArg(z1)), map), ENCARG(cons_app(z0, z1)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(ack, map)) -> c10(APP(ack, map), ENCARG(ack))
   ENCARG(cons_app(0, map)) -> c10(APP(0, map), ENCARG(0))
   ENCARG(cons_app(succ, map)) -> c10(APP(succ, map), ENCARG(succ))
   ENCARG(cons_app(map, map)) -> c10(APP(map, map), ENCARG(map))
   ENCARG(cons_app(nil, map)) -> c10(APP(nil, map), ENCARG(nil))
   ENCARG(cons_app(cons, map)) -> c10(APP(cons, map), ENCARG(cons))
   ENCARG(cons_app(filter, map)) -> c10(APP(filter, map), ENCARG(filter))
   ENCARG(cons_app(filter2, map)) -> c10(APP(filter2, map), ENCARG(filter2))
   ENCARG(cons_app(true, map)) -> c10(APP(true, map), ENCARG(true))
   ENCARG(cons_app(false, map)) -> c10(APP(false, map), ENCARG(false))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCARG(cons_app(0, map)) -> c10(APP(0, map), ENCARG(0))
   ENCARG(cons_app(filter2, map)) -> c10(APP(filter2, map), ENCARG(filter2))
   ENCARG(cons_app(map, map)) -> c10(APP(map, map), ENCARG(map))
   ENCARG(cons_app(succ, map)) -> c10(APP(succ, map), ENCARG(succ))
   ENCARG(cons_app(true, map)) -> c10(APP(true, map), ENCARG(true))
   ENCARG(cons_app(cons, map)) -> c10(APP(cons, map), ENCARG(cons))
   ENCARG(cons_app(nil, map)) -> c10(APP(nil, map), ENCARG(nil))
   ENCARG(cons_app(false, map)) -> c10(APP(false, map), ENCARG(false))
   ENCARG(cons_app(filter, map)) -> c10(APP(filter, map), ENCARG(filter))
   ENCARG(cons_app(ack, map)) -> c10(APP(ack, map), ENCARG(ack))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, nil)) -> c10(APP(encArg(x0), nil), ENCARG(x0)) by 
   ENCARG(cons_app(ack, nil)) -> c10(APP(ack, nil), ENCARG(ack))
   ENCARG(cons_app(0, nil)) -> c10(APP(0, nil), ENCARG(0))
   ENCARG(cons_app(succ, nil)) -> c10(APP(succ, nil), ENCARG(succ))
   ENCARG(cons_app(map, nil)) -> c10(APP(map, nil), ENCARG(map))
   ENCARG(cons_app(nil, nil)) -> c10(APP(nil, nil), ENCARG(nil))
   ENCARG(cons_app(cons, nil)) -> c10(APP(cons, nil), ENCARG(cons))
   ENCARG(cons_app(filter, nil)) -> c10(APP(filter, nil), ENCARG(filter))
   ENCARG(cons_app(filter2, nil)) -> c10(APP(filter2, nil), ENCARG(filter2))
   ENCARG(cons_app(true, nil)) -> c10(APP(true, nil), ENCARG(true))
   ENCARG(cons_app(false, nil)) -> c10(APP(false, nil), ENCARG(false))
   ENCARG(cons_app(cons_app(z0, z1), nil)) -> c10(APP(app(encArg(z0), encArg(z1)), nil), ENCARG(cons_app(z0, z1)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(ack, nil)) -> c10(APP(ack, nil), ENCARG(ack))
   ENCARG(cons_app(0, nil)) -> c10(APP(0, nil), ENCARG(0))
   ENCARG(cons_app(succ, nil)) -> c10(APP(succ, nil), ENCARG(succ))
   ENCARG(cons_app(map, nil)) -> c10(APP(map, nil), ENCARG(map))
   ENCARG(cons_app(nil, nil)) -> c10(APP(nil, nil), ENCARG(nil))
   ENCARG(cons_app(cons, nil)) -> c10(APP(cons, nil), ENCARG(cons))
   ENCARG(cons_app(filter, nil)) -> c10(APP(filter, nil), ENCARG(filter))
   ENCARG(cons_app(filter2, nil)) -> c10(APP(filter2, nil), ENCARG(filter2))
   ENCARG(cons_app(true, nil)) -> c10(APP(true, nil), ENCARG(true))
   ENCARG(cons_app(false, nil)) -> c10(APP(false, nil), ENCARG(false))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCARG(cons_app(filter, nil)) -> c10(APP(filter, nil), ENCARG(filter))
   ENCARG(cons_app(cons, nil)) -> c10(APP(cons, nil), ENCARG(cons))
   ENCARG(cons_app(nil, nil)) -> c10(APP(nil, nil), ENCARG(nil))
   ENCARG(cons_app(true, nil)) -> c10(APP(true, nil), ENCARG(true))
   ENCARG(cons_app(ack, nil)) -> c10(APP(ack, nil), ENCARG(ack))
   ENCARG(cons_app(false, nil)) -> c10(APP(false, nil), ENCARG(false))
   ENCARG(cons_app(0, nil)) -> c10(APP(0, nil), ENCARG(0))
   ENCARG(cons_app(succ, nil)) -> c10(APP(succ, nil), ENCARG(succ))
   ENCARG(cons_app(filter2, nil)) -> c10(APP(filter2, nil), ENCARG(filter2))
   ENCARG(cons_app(map, nil)) -> c10(APP(map, nil), ENCARG(map))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, cons)) -> c10(APP(encArg(x0), cons), ENCARG(x0)) by 
   ENCARG(cons_app(ack, cons)) -> c10(APP(ack, cons), ENCARG(ack))
   ENCARG(cons_app(0, cons)) -> c10(APP(0, cons), ENCARG(0))
   ENCARG(cons_app(succ, cons)) -> c10(APP(succ, cons), ENCARG(succ))
   ENCARG(cons_app(map, cons)) -> c10(APP(map, cons), ENCARG(map))
   ENCARG(cons_app(nil, cons)) -> c10(APP(nil, cons), ENCARG(nil))
   ENCARG(cons_app(cons, cons)) -> c10(APP(cons, cons), ENCARG(cons))
   ENCARG(cons_app(filter, cons)) -> c10(APP(filter, cons), ENCARG(filter))
   ENCARG(cons_app(filter2, cons)) -> c10(APP(filter2, cons), ENCARG(filter2))
   ENCARG(cons_app(true, cons)) -> c10(APP(true, cons), ENCARG(true))
   ENCARG(cons_app(false, cons)) -> c10(APP(false, cons), ENCARG(false))
   ENCARG(cons_app(cons_app(z0, z1), cons)) -> c10(APP(app(encArg(z0), encArg(z1)), cons), ENCARG(cons_app(z0, z1)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(ack, cons)) -> c10(APP(ack, cons), ENCARG(ack))
   ENCARG(cons_app(0, cons)) -> c10(APP(0, cons), ENCARG(0))
   ENCARG(cons_app(succ, cons)) -> c10(APP(succ, cons), ENCARG(succ))
   ENCARG(cons_app(map, cons)) -> c10(APP(map, cons), ENCARG(map))
   ENCARG(cons_app(nil, cons)) -> c10(APP(nil, cons), ENCARG(nil))
   ENCARG(cons_app(cons, cons)) -> c10(APP(cons, cons), ENCARG(cons))
   ENCARG(cons_app(filter, cons)) -> c10(APP(filter, cons), ENCARG(filter))
   ENCARG(cons_app(filter2, cons)) -> c10(APP(filter2, cons), ENCARG(filter2))
   ENCARG(cons_app(true, cons)) -> c10(APP(true, cons), ENCARG(true))
   ENCARG(cons_app(false, cons)) -> c10(APP(false, cons), ENCARG(false))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCARG(cons_app(succ, cons)) -> c10(APP(succ, cons), ENCARG(succ))
   ENCARG(cons_app(0, cons)) -> c10(APP(0, cons), ENCARG(0))
   ENCARG(cons_app(cons, cons)) -> c10(APP(cons, cons), ENCARG(cons))
   ENCARG(cons_app(filter2, cons)) -> c10(APP(filter2, cons), ENCARG(filter2))
   ENCARG(cons_app(map, cons)) -> c10(APP(map, cons), ENCARG(map))
   ENCARG(cons_app(nil, cons)) -> c10(APP(nil, cons), ENCARG(nil))
   ENCARG(cons_app(filter, cons)) -> c10(APP(filter, cons), ENCARG(filter))
   ENCARG(cons_app(true, cons)) -> c10(APP(true, cons), ENCARG(true))
   ENCARG(cons_app(ack, cons)) -> c10(APP(ack, cons), ENCARG(ack))
   ENCARG(cons_app(false, cons)) -> c10(APP(false, cons), ENCARG(false))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, filter)) -> c10(APP(encArg(x0), filter), ENCARG(x0)) by 
   ENCARG(cons_app(ack, filter)) -> c10(APP(ack, filter), ENCARG(ack))
   ENCARG(cons_app(0, filter)) -> c10(APP(0, filter), ENCARG(0))
   ENCARG(cons_app(succ, filter)) -> c10(APP(succ, filter), ENCARG(succ))
   ENCARG(cons_app(map, filter)) -> c10(APP(map, filter), ENCARG(map))
   ENCARG(cons_app(nil, filter)) -> c10(APP(nil, filter), ENCARG(nil))
   ENCARG(cons_app(cons, filter)) -> c10(APP(cons, filter), ENCARG(cons))
   ENCARG(cons_app(filter, filter)) -> c10(APP(filter, filter), ENCARG(filter))
   ENCARG(cons_app(filter2, filter)) -> c10(APP(filter2, filter), ENCARG(filter2))
   ENCARG(cons_app(true, filter)) -> c10(APP(true, filter), ENCARG(true))
   ENCARG(cons_app(false, filter)) -> c10(APP(false, filter), ENCARG(false))
   ENCARG(cons_app(cons_app(z0, z1), filter)) -> c10(APP(app(encArg(z0), encArg(z1)), filter), ENCARG(cons_app(z0, z1)))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(ack, filter)) -> c10(APP(ack, filter), ENCARG(ack))
   ENCARG(cons_app(0, filter)) -> c10(APP(0, filter), ENCARG(0))
   ENCARG(cons_app(succ, filter)) -> c10(APP(succ, filter), ENCARG(succ))
   ENCARG(cons_app(map, filter)) -> c10(APP(map, filter), ENCARG(map))
   ENCARG(cons_app(nil, filter)) -> c10(APP(nil, filter), ENCARG(nil))
   ENCARG(cons_app(cons, filter)) -> c10(APP(cons, filter), ENCARG(cons))
   ENCARG(cons_app(filter, filter)) -> c10(APP(filter, filter), ENCARG(filter))
   ENCARG(cons_app(filter2, filter)) -> c10(APP(filter2, filter), ENCARG(filter2))
   ENCARG(cons_app(true, filter)) -> c10(APP(true, filter), ENCARG(true))
   ENCARG(cons_app(false, filter)) -> c10(APP(false, filter), ENCARG(false))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCARG(cons_app(true, filter)) -> c10(APP(true, filter), ENCARG(true))
   ENCARG(cons_app(succ, filter)) -> c10(APP(succ, filter), ENCARG(succ))
   ENCARG(cons_app(nil, filter)) -> c10(APP(nil, filter), ENCARG(nil))
   ENCARG(cons_app(cons, filter)) -> c10(APP(cons, filter), ENCARG(cons))
   ENCARG(cons_app(filter, filter)) -> c10(APP(filter, filter), ENCARG(filter))
   ENCARG(cons_app(ack, filter)) -> c10(APP(ack, filter), ENCARG(ack))
   ENCARG(cons_app(filter2, filter)) -> c10(APP(filter2, filter), ENCARG(filter2))
   ENCARG(cons_app(map, filter)) -> c10(APP(map, filter), ENCARG(map))
   ENCARG(cons_app(false, filter)) -> c10(APP(false, filter), ENCARG(false))
   ENCARG(cons_app(0, filter)) -> c10(APP(0, filter), ENCARG(0))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, filter2)) -> c10(APP(encArg(x0), filter2), ENCARG(x0)) by 
   ENCARG(cons_app(ack, filter2)) -> c10(APP(ack, filter2), ENCARG(ack))
   ENCARG(cons_app(0, filter2)) -> c10(APP(0, filter2), ENCARG(0))
   ENCARG(cons_app(succ, filter2)) -> c10(APP(succ, filter2), ENCARG(succ))
   ENCARG(cons_app(map, filter2)) -> c10(APP(map, filter2), ENCARG(map))
   ENCARG(cons_app(nil, filter2)) -> c10(APP(nil, filter2), ENCARG(nil))
   ENCARG(cons_app(cons, filter2)) -> c10(APP(cons, filter2), ENCARG(cons))
   ENCARG(cons_app(filter, filter2)) -> c10(APP(filter, filter2), ENCARG(filter))
   ENCARG(cons_app(filter2, filter2)) -> c10(APP(filter2, filter2), ENCARG(filter2))
   ENCARG(cons_app(true, filter2)) -> c10(APP(true, filter2), ENCARG(true))
   ENCARG(cons_app(false, filter2)) -> c10(APP(false, filter2), ENCARG(false))
   ENCARG(cons_app(cons_app(z0, z1), filter2)) -> c10(APP(app(encArg(z0), encArg(z1)), filter2), ENCARG(cons_app(z0, z1)))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(ack, filter2)) -> c10(APP(ack, filter2), ENCARG(ack))
   ENCARG(cons_app(0, filter2)) -> c10(APP(0, filter2), ENCARG(0))
   ENCARG(cons_app(succ, filter2)) -> c10(APP(succ, filter2), ENCARG(succ))
   ENCARG(cons_app(map, filter2)) -> c10(APP(map, filter2), ENCARG(map))
   ENCARG(cons_app(nil, filter2)) -> c10(APP(nil, filter2), ENCARG(nil))
   ENCARG(cons_app(cons, filter2)) -> c10(APP(cons, filter2), ENCARG(cons))
   ENCARG(cons_app(filter, filter2)) -> c10(APP(filter, filter2), ENCARG(filter))
   ENCARG(cons_app(filter2, filter2)) -> c10(APP(filter2, filter2), ENCARG(filter2))
   ENCARG(cons_app(true, filter2)) -> c10(APP(true, filter2), ENCARG(true))
   ENCARG(cons_app(false, filter2)) -> c10(APP(false, filter2), ENCARG(false))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCARG(cons_app(nil, filter2)) -> c10(APP(nil, filter2), ENCARG(nil))
   ENCARG(cons_app(succ, filter2)) -> c10(APP(succ, filter2), ENCARG(succ))
   ENCARG(cons_app(filter, filter2)) -> c10(APP(filter, filter2), ENCARG(filter))
   ENCARG(cons_app(0, filter2)) -> c10(APP(0, filter2), ENCARG(0))
   ENCARG(cons_app(cons, filter2)) -> c10(APP(cons, filter2), ENCARG(cons))
   ENCARG(cons_app(ack, filter2)) -> c10(APP(ack, filter2), ENCARG(ack))
   ENCARG(cons_app(false, filter2)) -> c10(APP(false, filter2), ENCARG(false))
   ENCARG(cons_app(filter2, filter2)) -> c10(APP(filter2, filter2), ENCARG(filter2))
   ENCARG(cons_app(map, filter2)) -> c10(APP(map, filter2), ENCARG(map))
   ENCARG(cons_app(true, filter2)) -> c10(APP(true, filter2), ENCARG(true))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, true)) -> c10(APP(encArg(x0), true), ENCARG(x0)) by 
   ENCARG(cons_app(ack, true)) -> c10(APP(ack, true), ENCARG(ack))
   ENCARG(cons_app(0, true)) -> c10(APP(0, true), ENCARG(0))
   ENCARG(cons_app(succ, true)) -> c10(APP(succ, true), ENCARG(succ))
   ENCARG(cons_app(map, true)) -> c10(APP(map, true), ENCARG(map))
   ENCARG(cons_app(nil, true)) -> c10(APP(nil, true), ENCARG(nil))
   ENCARG(cons_app(cons, true)) -> c10(APP(cons, true), ENCARG(cons))
   ENCARG(cons_app(filter, true)) -> c10(APP(filter, true), ENCARG(filter))
   ENCARG(cons_app(filter2, true)) -> c10(APP(filter2, true), ENCARG(filter2))
   ENCARG(cons_app(true, true)) -> c10(APP(true, true), ENCARG(true))
   ENCARG(cons_app(false, true)) -> c10(APP(false, true), ENCARG(false))
   ENCARG(cons_app(cons_app(z0, z1), true)) -> c10(APP(app(encArg(z0), encArg(z1)), true), ENCARG(cons_app(z0, z1)))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(ack, true)) -> c10(APP(ack, true), ENCARG(ack))
   ENCARG(cons_app(0, true)) -> c10(APP(0, true), ENCARG(0))
   ENCARG(cons_app(succ, true)) -> c10(APP(succ, true), ENCARG(succ))
   ENCARG(cons_app(map, true)) -> c10(APP(map, true), ENCARG(map))
   ENCARG(cons_app(nil, true)) -> c10(APP(nil, true), ENCARG(nil))
   ENCARG(cons_app(cons, true)) -> c10(APP(cons, true), ENCARG(cons))
   ENCARG(cons_app(filter, true)) -> c10(APP(filter, true), ENCARG(filter))
   ENCARG(cons_app(filter2, true)) -> c10(APP(filter2, true), ENCARG(filter2))
   ENCARG(cons_app(true, true)) -> c10(APP(true, true), ENCARG(true))
   ENCARG(cons_app(false, true)) -> c10(APP(false, true), ENCARG(false))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCARG(cons_app(nil, true)) -> c10(APP(nil, true), ENCARG(nil))
   ENCARG(cons_app(succ, true)) -> c10(APP(succ, true), ENCARG(succ))
   ENCARG(cons_app(0, true)) -> c10(APP(0, true), ENCARG(0))
   ENCARG(cons_app(filter, true)) -> c10(APP(filter, true), ENCARG(filter))
   ENCARG(cons_app(ack, true)) -> c10(APP(ack, true), ENCARG(ack))
   ENCARG(cons_app(filter2, true)) -> c10(APP(filter2, true), ENCARG(filter2))
   ENCARG(cons_app(map, true)) -> c10(APP(map, true), ENCARG(map))
   ENCARG(cons_app(false, true)) -> c10(APP(false, true), ENCARG(false))
   ENCARG(cons_app(true, true)) -> c10(APP(true, true), ENCARG(true))
   ENCARG(cons_app(cons, true)) -> c10(APP(cons, true), ENCARG(cons))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_2, c10_1, c24_2, c28_2, c28_1, c_1, c10_3


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

(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_app(x0, false)) -> c10(APP(encArg(x0), false), ENCARG(x0)) by 
   ENCARG(cons_app(ack, false)) -> c10(APP(ack, false), ENCARG(ack))
   ENCARG(cons_app(0, false)) -> c10(APP(0, false), ENCARG(0))
   ENCARG(cons_app(succ, false)) -> c10(APP(succ, false), ENCARG(succ))
   ENCARG(cons_app(map, false)) -> c10(APP(map, false), ENCARG(map))
   ENCARG(cons_app(nil, false)) -> c10(APP(nil, false), ENCARG(nil))
   ENCARG(cons_app(cons, false)) -> c10(APP(cons, false), ENCARG(cons))
   ENCARG(cons_app(filter, false)) -> c10(APP(filter, false), ENCARG(filter))
   ENCARG(cons_app(filter2, false)) -> c10(APP(filter2, false), ENCARG(filter2))
   ENCARG(cons_app(true, false)) -> c10(APP(true, false), ENCARG(true))
   ENCARG(cons_app(false, false)) -> c10(APP(false, false), ENCARG(false))
   ENCARG(cons_app(cons_app(z0, z1), false)) -> c10(APP(app(encArg(z0), encArg(z1)), false), ENCARG(cons_app(z0, z1)))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   ENCARG(cons_app(ack, false)) -> c10(APP(ack, false), ENCARG(ack))
   ENCARG(cons_app(0, false)) -> c10(APP(0, false), ENCARG(0))
   ENCARG(cons_app(succ, false)) -> c10(APP(succ, false), ENCARG(succ))
   ENCARG(cons_app(map, false)) -> c10(APP(map, false), ENCARG(map))
   ENCARG(cons_app(nil, false)) -> c10(APP(nil, false), ENCARG(nil))
   ENCARG(cons_app(cons, false)) -> c10(APP(cons, false), ENCARG(cons))
   ENCARG(cons_app(filter, false)) -> c10(APP(filter, false), ENCARG(filter))
   ENCARG(cons_app(filter2, false)) -> c10(APP(filter2, false), ENCARG(filter2))
   ENCARG(cons_app(true, false)) -> c10(APP(true, false), ENCARG(true))
   ENCARG(cons_app(false, false)) -> c10(APP(false, false), ENCARG(false))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_1, c24_2, c28_2, c28_1, c_1, c10_3, c10_2


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

(107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   ENCARG(cons_app(cons, false)) -> c10(APP(cons, false), ENCARG(cons))
   ENCARG(cons_app(nil, false)) -> c10(APP(nil, false), ENCARG(nil))
   ENCARG(cons_app(false, false)) -> c10(APP(false, false), ENCARG(false))
   ENCARG(cons_app(ack, false)) -> c10(APP(ack, false), ENCARG(ack))
   ENCARG(cons_app(true, false)) -> c10(APP(true, false), ENCARG(true))
   ENCARG(cons_app(filter, false)) -> c10(APP(filter, false), ENCARG(filter))
   ENCARG(cons_app(0, false)) -> c10(APP(0, false), ENCARG(0))
   ENCARG(cons_app(succ, false)) -> c10(APP(succ, false), ENCARG(succ))
   ENCARG(cons_app(filter2, false)) -> c10(APP(filter2, false), ENCARG(filter2))
   ENCARG(cons_app(map, false)) -> c10(APP(map, false), ENCARG(map))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_1, c24_2, c28_2, c28_1, c_1, c10_3, c10_2


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APP(app(ack, app(succ, z0)), app(succ, z1)) -> c24(APP(app(ack, z0), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, z0)), z1)) by 
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_1, c24_2, c28_2, c28_1, c_1, c10_3, c10_2, c24_1


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), APP(app(ack, app(succ, z0)), app(succ, z1))) by 
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_1, c28_2, c28_1, c_1, c10_3, c10_2, c24_2, c24_1


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APP(app(filter, app(ack, app(succ, z0))), app(app(cons, z1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, z0))), z1), x2), APP(app(ack, app(succ, z0)), z1)) by 
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(succ, app(succ, 0))), app(ack, app(succ, 0))), x1), x2), APP(app(ack, app(succ, 0)), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, x0)), x1))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(succ, app(succ, 0))), app(ack, app(succ, 0))), x1), x2), APP(app(ack, app(succ, 0)), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, x0)), x1))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(succ, app(succ, 0))), app(ack, app(succ, 0))), x1), x2), APP(app(ack, app(succ, 0)), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, x0)), x1))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_1, c28_2, c28_1, c_1, c10_3, c10_2, c24_2, c24_1


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

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, x0)), x1))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, 0)), x1))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, x0)), x1))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, 0)), x1))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_1, c28_2, c28_1, c_1, c10_3, c10_2, c24_2, c24_1


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), z1))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1))) by 
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(app(app(filter2, app(succ, app(app(ack, app(succ, 0)), x1))), app(ack, app(succ, 0))), app(succ, x1)), x2), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), app(succ, x1)), x2), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, app(succ, z1))), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), app(ack, app(succ, z0))), app(succ, app(succ, z1))), x2), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(ack, app(succ, x0)), app(succ, x1)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, x0)), x1))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, 0)), x1))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(app(app(filter2, app(succ, app(app(ack, app(succ, 0)), x1))), app(ack, app(succ, 0))), app(succ, x1)), x2), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), app(succ, x1)), x2), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, app(succ, z1))), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), app(ack, app(succ, z0))), app(succ, app(succ, z1))), x2), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(ack, app(succ, x0)), app(succ, x1)))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, x0)), x1))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, 0)), x1))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(app(app(filter2, app(succ, app(app(ack, app(succ, 0)), x1))), app(ack, app(succ, 0))), app(succ, x1)), x2), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), app(succ, x1)), x2), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, app(succ, z1))), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), app(ack, app(succ, z0))), app(succ, app(succ, z1))), x2), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(ack, app(succ, x0)), app(succ, x1)))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_1, c28_2, c28_1, c_1, c10_3, c10_2, c24_2, c24_1


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

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(ack) -> ack
   encArg(0) -> 0
   encArg(succ) -> succ
   encArg(map) -> map
   encArg(nil) -> nil
   encArg(cons) -> cons
   encArg(filter) -> filter
   encArg(filter2) -> filter2
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(app(ack, 0), z0) -> app(succ, z0)
   app(app(ack, app(succ, z0)), z1) -> app(app(ack, z0), app(succ, 0))
   app(app(ack, app(succ, z0)), app(succ, z1)) -> app(app(ack, z0), app(app(ack, app(succ, z0)), z1))
   app(app(map, z0), nil) -> nil
   app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2))
   app(app(filter, z0), nil) -> nil
   app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)
   app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2))
   app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2)
Tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   ENCARG(cons_app(ack, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(0, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(succ, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(map, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(nil, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(cons, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(filter2, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(true, x1)) -> c10(ENCARG(x1))
   ENCARG(cons_app(false, x1)) -> c10(ENCARG(x1))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   ENCODE_APP(x0, ack) -> c(APP(encArg(x0), ack))
   ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0))
   ENCODE_APP(x0, succ) -> c(APP(encArg(x0), succ))
   ENCODE_APP(x0, map) -> c(APP(encArg(x0), map))
   ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil))
   ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons))
   ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter))
   ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2))
   ENCODE_APP(x0, true) -> c(APP(encArg(x0), true))
   ENCODE_APP(x0, false) -> c(APP(encArg(x0), false))
   ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1))))
   ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_app(x0, cons_app(x1, ack))) -> c10(APP(encArg(x0), app(encArg(x1), ack)), ENCARG(x0), ENCARG(cons_app(x1, ack)))
   ENCARG(cons_app(x0, cons_app(x1, 0))) -> c10(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0)))
   ENCARG(cons_app(x0, cons_app(x1, succ))) -> c10(APP(encArg(x0), app(encArg(x1), succ)), ENCARG(x0), ENCARG(cons_app(x1, succ)))
   ENCARG(cons_app(x0, cons_app(x1, map))) -> c10(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map)))
   ENCARG(cons_app(x0, cons_app(x1, nil))) -> c10(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil)))
   ENCARG(cons_app(x0, cons_app(x1, cons))) -> c10(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons)))
   ENCARG(cons_app(x0, cons_app(x1, filter))) -> c10(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter)))
   ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c10(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2)))
   ENCARG(cons_app(x0, cons_app(x1, true))) -> c10(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true)))
   ENCARG(cons_app(x0, cons_app(x1, false))) -> c10(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false)))
   ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c10(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1))))
   ENCARG(cons_app(x0, cons_app(ack, x2))) -> c10(APP(encArg(x0), app(ack, encArg(x2))), ENCARG(x0), ENCARG(cons_app(ack, x2)))
   ENCARG(cons_app(x0, cons_app(0, x2))) -> c10(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2)))
   ENCARG(cons_app(x0, cons_app(succ, x2))) -> c10(APP(encArg(x0), app(succ, encArg(x2))), ENCARG(x0), ENCARG(cons_app(succ, x2)))
   ENCARG(cons_app(x0, cons_app(map, x2))) -> c10(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2)))
   ENCARG(cons_app(x0, cons_app(nil, x2))) -> c10(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2)))
   ENCARG(cons_app(x0, cons_app(cons, x2))) -> c10(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2)))
   ENCARG(cons_app(x0, cons_app(filter, x2))) -> c10(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2)))
   ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c10(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2)))
   ENCARG(cons_app(x0, cons_app(true, x2))) -> c10(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2)))
   ENCARG(cons_app(x0, cons_app(false, x2))) -> c10(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2)))
   ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c10(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2)))
   ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c10(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(ack, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(0, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(succ, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(map, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(nil, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(true, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(false, cons_app(x1, x2))) -> c10(ENCARG(cons_app(x1, x2)))
   ENCARG(cons_app(cons_app(x0, ack), x2)) -> c10(APP(app(encArg(x0), ack), encArg(x2)), ENCARG(cons_app(x0, ack)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, 0), x2)) -> c10(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, succ), x2)) -> c10(APP(app(encArg(x0), succ), encArg(x2)), ENCARG(cons_app(x0, succ)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, map), x2)) -> c10(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, nil), x2)) -> c10(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons), x2)) -> c10(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter), x2)) -> c10(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c10(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, true), x2)) -> c10(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, false), x2)) -> c10(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c10(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2))
   ENCARG(cons_app(cons_app(ack, x1), x2)) -> c10(APP(app(ack, encArg(x1)), encArg(x2)), ENCARG(cons_app(ack, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(map, x1), x2)) -> c10(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter, x1), x2)) -> c10(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c10(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(x0, x1), ack)) -> c10(APP(app(encArg(x0), encArg(x1)), ack), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), 0)) -> c10(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), succ)) -> c10(APP(app(encArg(x0), encArg(x1)), succ), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), map)) -> c10(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), nil)) -> c10(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), cons)) -> c10(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter)) -> c10(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c10(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), true)) -> c10(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(x0, x1), false)) -> c10(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)))
   ENCARG(cons_app(cons_app(0, x1), x2)) -> c10(ENCARG(cons_app(0, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(succ, x1), x2)) -> c10(ENCARG(cons_app(succ, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(nil, x1), x2)) -> c10(ENCARG(cons_app(nil, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(cons, x1), x2)) -> c10(ENCARG(cons_app(cons, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c10(ENCARG(cons_app(filter2, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(true, x1), x2)) -> c10(ENCARG(cons_app(true, x1)), ENCARG(x2))
   ENCARG(cons_app(cons_app(false, x1), x2)) -> c10(ENCARG(cons_app(false, x1)), ENCARG(x2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, x0)), x1))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, 0)), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), app(succ, x1)), x2), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, app(succ, z1))), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), app(ack, app(succ, z0))), app(succ, app(succ, z1))), x2), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(ack, app(succ, x0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(ack, app(succ, 0)), app(succ, x1)))
S tuples:
   APP(app(map, z0), nil) -> c25
   APP(app(filter, z0), nil) -> c27
   APP(app(ack, 0), z0) -> c22
   APP(app(ack, app(succ, z0)), z1) -> c23(APP(app(ack, z0), app(succ, 0)))
   APP(app(map, z0), app(app(cons, z1), z2)) -> c26(APP(z0, z1), APP(app(map, z0), z2))
   APP(app(app(app(filter2, true), z0), z1), z2) -> c29(APP(app(filter, z0), z2))
   APP(app(app(app(filter2, false), z0), z1), z2) -> c30(APP(app(filter, z0), z2))
   APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2))
   APP(app(filter, app(ack, 0)), app(app(cons, z0), x2)) -> c28(APP(app(ack, 0), z0))
   APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c28(APP(app(map, z0), nil))
   APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c28(APP(app(map, z0), app(app(cons, z1), z2)))
   APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c28(APP(app(filter, z0), nil))
   APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c28(APP(app(app(app(filter2, true), z0), z1), z2))
   APP(app(ack, app(succ, 0)), app(succ, x1)) -> c24(APP(app(ack, 0), app(succ, app(succ, 0))), APP(app(ack, app(succ, 0)), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(ack, app(succ, x0)), app(succ, x1)) -> c24(APP(app(ack, app(succ, x0)), x1))
   APP(app(ack, app(succ, 0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), x1))), APP(app(ack, app(succ, 0)), app(succ, x1)))
   APP(app(ack, app(succ, app(succ, z0))), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, z0)), app(app(ack, z0), app(succ, 0))), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(ack, app(succ, z0)), app(succ, app(succ, app(succ, z1)))) -> c24(APP(app(ack, z0), app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(ack, app(succ, x0)), app(succ, app(succ, x1))) -> c24(APP(app(ack, app(succ, x0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, x1), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, app(succ, z0)), 0))), app(ack, app(succ, app(succ, z0)))), x1), x2), APP(app(ack, app(succ, app(succ, z0))), x1))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, x0)), x1))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, x1), x2)) -> c28(APP(app(ack, app(succ, 0)), x1))
   APP(app(filter, app(ack, app(succ, app(succ, z0)))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(succ, 0))), app(ack, app(succ, app(succ, z0)))), app(succ, x1)), x2), APP(app(ack, app(succ, app(succ, z0))), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, z1)), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, z0), app(succ, 0)))), app(ack, app(succ, z0))), app(succ, z1)), x2), APP(app(ack, app(succ, z0)), app(succ, z1)))
   APP(app(filter, app(ack, app(succ, z0))), app(app(cons, app(succ, app(succ, z1))), x2)) -> c28(APP(app(app(app(filter2, app(app(ack, z0), app(app(ack, z0), app(app(ack, app(succ, z0)), z1)))), app(ack, app(succ, z0))), app(succ, app(succ, z1))), x2), APP(app(ack, app(succ, z0)), app(succ, app(succ, z1))))
   APP(app(filter, app(ack, app(succ, x0))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(ack, app(succ, x0)), app(succ, x1)))
   APP(app(filter, app(ack, app(succ, 0))), app(app(cons, app(succ, x1)), x2)) -> c28(APP(app(ack, app(succ, 0)), app(succ, x1)))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   APP_2, ENCARG_1, ENCODE_APP_2

Compound Symbols:   c25, c27, c22, c23_1, c26_2, c29_1, c30_1, c10_1, c28_2, c28_1, c_1, c10_3, c10_2, c24_2, c24_1