KILLED
proof of input_BUIWobVp7S.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 384 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 90 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 215 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 49 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 958 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 446 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 1037 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 376 ms]
        (42) CpxRNTS
        (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 1508 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 518 ms]
        (48) CpxRNTS
    (49) CompletionProof [UPPER BOUND(ID), 0 ms]
    (50) CpxTypedWeightedCompleteTrs
        (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
(53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(54) CdtProblem
    (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 156 ms]
    (62) CdtProblem
    (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
    (94) CdtProblem
    (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms]
    (172) CdtProblem
    (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem


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

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


The TRS R consists of the following rules:

   qsort(nil) -> nil
   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs)))))
   filterlow(n, nil) -> nil
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
   if1(true, n, x, xs) -> filterlow(n, xs)
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
   filterhigh(n, nil) -> nil
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
   if2(true, n, x, xs) -> filterhigh(n, xs)
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   append(nil, ys) -> ys
   append(cons(x, xs), ys) -> cons(x, append(xs, ys))

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

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

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


The TRS R consists of the following rules:

   qsort(nil) -> nil
   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs)))))
   filterlow(n, nil) -> nil
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
   if1(true, n, x, xs) -> filterlow(n, xs)
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
   filterhigh(n, nil) -> nil
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
   if2(true, n, x, xs) -> filterhigh(n, xs)
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   append(nil, ys) -> ys
   append(cons(x, xs), ys) -> cons(x, append(xs, ys))

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

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

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


The TRS R consists of the following rules:

   qsort(nil) -> nil
   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs)))))
   filterlow(n, nil) -> nil
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
   if1(true, n, x, xs) -> filterlow(n, xs)
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
   filterhigh(n, nil) -> nil
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
   if2(true, n, x, xs) -> filterhigh(n, xs)
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   append(nil, ys) -> ys
   append(cons(x, xs), ys) -> cons(x, append(xs, ys))

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

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

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


The TRS R consists of the following rules:

   qsort(nil) -> nil [1]
   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs))))) [1]
   filterlow(n, nil) -> nil [1]
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1]
   if1(true, n, x, xs) -> filterlow(n, xs) [1]
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1]
   filterhigh(n, nil) -> nil [1]
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1]
   if2(true, n, x, xs) -> filterhigh(n, xs) [1]
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   append(nil, ys) -> ys [1]
   append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   qsort(nil) -> nil [1]
   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs))))) [1]
   filterlow(n, nil) -> nil [1]
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1]
   if1(true, n, x, xs) -> filterlow(n, xs) [1]
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1]
   filterhigh(n, nil) -> nil [1]
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1]
   if2(true, n, x, xs) -> filterhigh(n, xs) [1]
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   append(nil, ys) -> ys [1]
   append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1]

The TRS has the following type information:
qsort :: nil:cons:ys -> nil:cons:ys
nil :: nil:cons:ys
cons :: 0:s -> nil:cons:ys -> nil:cons:ys
append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys
filterlow :: 0:s -> nil:cons:ys -> nil:cons:ys
filterhigh :: 0:s -> nil:cons:ys -> nil:cons:ys
if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys -> nil:cons:ys
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys -> nil:cons:ys
0 :: 0:s
s :: 0:s -> 0:s
ys :: nil:cons:ys

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

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   qsort_1
   filterlow_2
   filterhigh_2
   ge_2
   append_2
   if1_4
   if2_4

Due to the following rules being added:

   qsort(v0) -> null_qsort [0]
   filterlow(v0, v1) -> null_filterlow [0]
   filterhigh(v0, v1) -> null_filterhigh [0]
   append(v0, v1) -> null_append [0]

And the following fresh constants:    null_qsort, null_filterlow, null_filterhigh, null_append

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

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

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

   qsort(nil) -> nil [1]
   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs))))) [1]
   filterlow(n, nil) -> nil [1]
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1]
   if1(true, n, x, xs) -> filterlow(n, xs) [1]
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1]
   filterhigh(n, nil) -> nil [1]
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1]
   if2(true, n, x, xs) -> filterhigh(n, xs) [1]
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   append(nil, ys) -> ys [1]
   append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1]
   qsort(v0) -> null_qsort [0]
   filterlow(v0, v1) -> null_filterlow [0]
   filterhigh(v0, v1) -> null_filterhigh [0]
   append(v0, v1) -> null_append [0]

The TRS has the following type information:
qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
nil :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
cons :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
filterlow :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
filterhigh :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
0 :: 0:s
s :: 0:s -> 0:s
ys :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_filterlow :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_filterhigh :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   qsort(nil) -> nil [1]
   qsort(cons(x, xs)) -> append(qsort(if1(ge(x, x), x, x, xs)), cons(x, qsort(if2(ge(x, x), x, x, xs)))) [3]
   qsort(cons(x, xs)) -> append(qsort(if1(ge(x, x), x, x, xs)), cons(x, qsort(null_filterhigh))) [2]
   qsort(cons(x, xs)) -> append(qsort(null_filterlow), cons(x, qsort(if2(ge(x, x), x, x, xs)))) [2]
   qsort(cons(x, xs)) -> append(qsort(null_filterlow), cons(x, qsort(null_filterhigh))) [1]
   filterlow(n, nil) -> nil [1]
   filterlow(n, cons(0, xs)) -> if1(true, n, 0, xs) [2]
   filterlow(0, cons(s(x'), xs)) -> if1(false, 0, s(x'), xs) [2]
   filterlow(s(x''), cons(s(y'), xs)) -> if1(ge(x'', y'), s(x''), s(y'), xs) [2]
   if1(true, n, x, xs) -> filterlow(n, xs) [1]
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1]
   filterhigh(n, nil) -> nil [1]
   filterhigh(0, cons(x, xs)) -> if2(true, 0, x, xs) [2]
   filterhigh(s(x1), cons(0, xs)) -> if2(false, s(x1), 0, xs) [2]
   filterhigh(s(y''), cons(s(x2), xs)) -> if2(ge(x2, y''), s(y''), s(x2), xs) [2]
   if2(true, n, x, xs) -> filterhigh(n, xs) [1]
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   append(nil, ys) -> ys [1]
   append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1]
   qsort(v0) -> null_qsort [0]
   filterlow(v0, v1) -> null_filterlow [0]
   filterhigh(v0, v1) -> null_filterhigh [0]
   append(v0, v1) -> null_append [0]

The TRS has the following type information:
qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
nil :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
cons :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
filterlow :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
filterhigh :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
0 :: 0:s
s :: 0:s -> 0:s
ys :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_filterlow :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_filterhigh :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append

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

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

   nil => 0
   true => 1
   false => 0
   0 => 0
   ys => 1
   null_qsort => 0
   null_filterlow => 0
   null_filterhigh => 0
   null_append => 0

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, y''), 1 + y'', 1 + x2, xs) :|: xs >= 0, z = 1 + y'', z' = 1 + (1 + x2) + xs, y'' >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + x1, 0, xs) :|: xs >= 0, x1 >= 0, z' = 1 + 0 + xs, z = 1 + x1
   filterhigh(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   filterlow(z, z') -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: z = 1 + x'', xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, x'' >= 0
   filterlow(z, z') -{ 2 }-> if1(1, n, 0, xs) :|: n >= 0, xs >= 0, z' = 1 + 0 + xs, z = n
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ge(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs
   if1(z, z', z'', z1) -{ 1 }-> 1 + x + filterlow(n, xs) :|: n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z = 0, z1 = xs
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs
   if2(z, z', z'', z1) -{ 1 }-> 1 + x + filterhigh(n, xs) :|: n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z = 0, z1 = xs
   qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { append }
   { ge }
   { filterlow, if1 }
   { filterhigh, if2 }
   { qsort }

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {append}, {ge}, {filterlow,if1}, {filterhigh,if2}, {qsort}

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {append}, {ge}, {filterlow,if1}, {filterhigh,if2}, {qsort}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {append}, {ge}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: ?, size: O(n^1) [1 + z]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ge}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ge}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ge}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: ?, size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: ge
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y'), 1 + (z - 1), 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 3 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(if1(ge(x, x), x, x, xs)), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 2 }-> append(qsort(0), 1 + x + qsort(if2(ge(x, x), x, x, xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 4 + y' }-> if1(s3, 1 + (z - 1), 1 + y', xs) :|: s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 7 + 2*x }-> append(qsort(if1(s', x, x, xs)), 1 + x + qsort(if2(s'', x, x, xs))) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 4 + x }-> append(qsort(if1(s1, x, x, xs)), 1 + x + qsort(0)) :|: s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: if1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'' + z1

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 4 + y' }-> if1(s3, 1 + (z - 1), 1 + y', xs) :|: s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 7 + 2*x }-> append(qsort(if1(s', x, x, xs)), 1 + x + qsort(if2(s'', x, x, xs))) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 4 + x }-> append(qsort(if1(s1, x, x, xs)), 1 + x + qsort(0)) :|: s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  filterlow: runtime: ?, size: O(n^1) [z']
  if1: runtime: ?, size: O(n^1) [1 + z'' + z1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: filterlow
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 14 + 7*z'

Computed RUNTIME bound using CoFloCo for: if1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 15 + 7*z1

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 4 + y' }-> if1(s3, 1 + (z - 1), 1 + y', xs) :|: s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 7 + 2*x }-> append(qsort(if1(s', x, x, xs)), 1 + x + qsort(if2(s'', x, x, xs))) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 4 + x }-> append(qsort(if1(s1, x, x, xs)), 1 + x + qsort(0)) :|: s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 22 + 2*x + 7*xs }-> append(qsort(s6), 1 + x + qsort(if2(s'', x, x, xs))) :|: s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'' + z1

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 22 + 2*x + 7*xs }-> append(qsort(s6), 1 + x + qsort(if2(s'', x, x, xs))) :|: s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]
  filterhigh: runtime: ?, size: O(n^1) [z']
  if2: runtime: ?, size: O(n^1) [1 + z'' + z1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: filterhigh
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 11 + 3*z + z*z' + 4*z'

Computed RUNTIME bound using KoAT for: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 24 + 6*z' + 2*z'*z1 + 8*z1

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s4, 1 + (z - 1), 1 + x2, xs) :|: s4 >= 0, s4 <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 22 + 2*x + 7*xs }-> append(qsort(s6), 1 + x + qsort(if2(s'', x, x, xs))) :|: s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 4 + x }-> append(qsort(0), 1 + x + qsort(if2(s2, x, x, xs))) :|: s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]
  filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1]

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 26 + 8*xs }-> s15 :|: s15 >= 0, s15 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s16 :|: s16 >= 0, s16 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s17 :|: s17 >= 0, s17 <= 1 + x2 + xs + 1, s4 >= 0, s4 <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s18 :|: s18 >= 0, s18 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s19 :|: s19 >= 0, s19 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 46 + 8*x + 2*x*xs + 15*xs }-> append(qsort(s6), 1 + x + qsort(s13)) :|: s13 >= 0, s13 <= x + xs + 1, s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 28 + 7*x + 2*x*xs + 8*xs }-> append(qsort(0), 1 + x + qsort(s14)) :|: s14 >= 0, s14 <= x + xs + 1, s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]
  filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 26 + 8*xs }-> s15 :|: s15 >= 0, s15 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s16 :|: s16 >= 0, s16 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s17 :|: s17 >= 0, s17 <= 1 + x2 + xs + 1, s4 >= 0, s4 <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s18 :|: s18 >= 0, s18 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s19 :|: s19 >= 0, s19 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 46 + 8*x + 2*x*xs + 15*xs }-> append(qsort(s6), 1 + x + qsort(s13)) :|: s13 >= 0, s13 <= x + xs + 1, s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 28 + 7*x + 2*x*xs + 8*xs }-> append(qsort(0), 1 + x + qsort(s14)) :|: s14 >= 0, s14 <= x + xs + 1, s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]
  filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1]
  qsort: runtime: ?, size: INF

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 26 + 8*xs }-> s15 :|: s15 >= 0, s15 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s16 :|: s16 >= 0, s16 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s17 :|: s17 >= 0, s17 <= 1 + x2 + xs + 1, s4 >= 0, s4 <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 19 + 7*xs + y' }-> s10 :|: s10 >= 0, s10 <= 1 + y' + xs + 1, s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s9 :|: s9 >= 0, s9 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s5 :|: s5 >= 0, s5 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s11 :|: s11 >= 0, s11 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s12 :|: s12 >= 0, s12 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s18 :|: s18 >= 0, s18 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s19 :|: s19 >= 0, s19 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   qsort(z) -{ 46 + 8*x + 2*x*xs + 15*xs }-> append(qsort(s6), 1 + x + qsort(s13)) :|: s13 >= 0, s13 <= x + xs + 1, s6 >= 0, s6 <= x + xs + 1, s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 19 + x + 7*xs }-> append(qsort(s7), 1 + x + qsort(0)) :|: s7 >= 0, s7 <= x + xs + 1, s1 >= 0, s1 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 28 + 7*x + 2*x*xs + 8*xs }-> append(qsort(0), 1 + x + qsort(s14)) :|: s14 >= 0, s14 <= x + xs + 1, s2 >= 0, s2 <= 1, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> append(qsort(0), 1 + x + qsort(0)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]
  filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1]
  qsort: runtime: INF, size: INF

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

(49) 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:

   qsort(v0) -> null_qsort [0]
   filterlow(v0, v1) -> null_filterlow [0]
   filterhigh(v0, v1) -> null_filterhigh [0]
   append(v0, v1) -> null_append [0]

And the following fresh constants:    null_qsort, null_filterlow, null_filterhigh, null_append

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

(50)
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:

   qsort(nil) -> nil [1]
   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs))))) [1]
   filterlow(n, nil) -> nil [1]
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1]
   if1(true, n, x, xs) -> filterlow(n, xs) [1]
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1]
   filterhigh(n, nil) -> nil [1]
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1]
   if2(true, n, x, xs) -> filterhigh(n, xs) [1]
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   append(nil, ys) -> ys [1]
   append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1]
   qsort(v0) -> null_qsort [0]
   filterlow(v0, v1) -> null_filterlow [0]
   filterhigh(v0, v1) -> null_filterhigh [0]
   append(v0, v1) -> null_append [0]

The TRS has the following type information:
qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
nil :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
cons :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
filterlow :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
filterhigh :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
0 :: 0:s
s :: 0:s -> 0:s
ys :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_filterlow :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_filterhigh :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append

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

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

   nil => 0
   true => 1
   false => 0
   0 => 0
   ys => 1
   null_qsort => 0
   null_filterlow => 0
   null_filterhigh => 0
   null_append => 0

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 1 }-> if2(ge(x, n), n, x, xs) :|: n >= 0, xs >= 0, z = n, z' = 1 + x + xs, x >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   filterlow(z, z') -{ 1 }-> if1(ge(n, x), n, x, xs) :|: n >= 0, xs >= 0, z = n, z' = 1 + x + xs, x >= 0
   filterlow(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ge(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs
   if1(z, z', z'', z1) -{ 1 }-> 1 + x + filterlow(n, xs) :|: n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z = 0, z1 = xs
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs
   if2(z, z', z'', z1) -{ 1 }-> 1 + x + filterhigh(n, xs) :|: n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z = 0, z1 = xs
   qsort(z) -{ 1 }-> append(qsort(filterlow(x, 1 + x + xs)), 1 + x + qsort(filterhigh(x, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   qsort(nil) -> nil
   qsort(cons(z0, z1)) -> append(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1)))))
   filterlow(z0, nil) -> nil
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, nil) -> nil
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   append(nil, ys) -> ys
   append(cons(z0, z1), ys) -> cons(z0, append(z1, ys))
Tuples:
   QSORT(nil) -> c
   QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(z0, nil) -> c3
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, nil) -> c7
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(z0, 0) -> c11
   GE(0, s(z0)) -> c12
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(nil, ys) -> c14
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
S tuples:
   QSORT(nil) -> c
   QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(z0, nil) -> c3
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, nil) -> c7
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(z0, 0) -> c11
   GE(0, s(z0)) -> c12
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(nil, ys) -> c14
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
K tuples:none
Defined Rule Symbols:   qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2

Defined Pair Symbols:   QSORT_1, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2

Compound Symbols:   c, c1_3, c2_3, c3, c4_2, c5_1, c6_1, c7, c8_2, c9_1, c10_1, c11, c12, c13_1, c14, c15_1


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

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   GE(0, s(z0)) -> c12
   FILTERHIGH(z0, nil) -> c7
   FILTERLOW(z0, nil) -> c3
   APPEND(nil, ys) -> c14
   GE(z0, 0) -> c11
   QSORT(nil) -> c

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   qsort(nil) -> nil
   qsort(cons(z0, z1)) -> append(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1)))))
   filterlow(z0, nil) -> nil
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, nil) -> nil
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   append(nil, ys) -> ys
   append(cons(z0, z1), ys) -> cons(z0, append(z1, ys))
Tuples:
   QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
S tuples:
   QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1))))), QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
K tuples:none
Defined Rule Symbols:   qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2

Defined Pair Symbols:   QSORT_1, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2

Compound Symbols:   c1_3, c2_3, c4_2, c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1


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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   qsort(nil) -> nil
   qsort(cons(z0, z1)) -> append(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1)))))
   filterlow(z0, nil) -> nil
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, nil) -> nil
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   append(nil, ys) -> ys
   append(cons(z0, z1), ys) -> cons(z0, append(z1, ys))
Tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
S tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2

Defined Pair Symbols:   FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, QSORT_1

Compound Symbols:   c4_2, c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2


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

(59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   qsort(nil) -> nil
   qsort(cons(z0, z1)) -> append(qsort(filterlow(z0, cons(z0, z1))), cons(z0, qsort(filterhigh(z0, cons(z0, z1)))))
   append(nil, ys) -> ys
   append(cons(z0, z1), ys) -> cons(z0, append(z1, ys))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
S tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, QSORT_1

Compound Symbols:   c4_2, c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2


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

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
We considered the (Usable) Rules:none
And the Tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(APPEND(x_1, x_2)) = x_1
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = 0
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(QSORT(x_1)) = 0
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = 0
   POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2
   POL(filterlow(x_1, x_2)) = [1] + x_1
   POL(ge(x_1, x_2)) = x_1 + x_2
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(nil) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = [1]
   POL(ys) = 0

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
S tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, QSORT_1

Compound Symbols:   c4_2, c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FILTERLOW(z0, cons(z1, z2)) -> c4(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) by 
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2), GE(z0, 0))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2), GE(0, s(z0)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2), GE(z0, 0))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2), GE(0, s(z0)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2), GE(z0, 0))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2), GE(0, s(z0)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2

Compound Symbols:   c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2, c4_2


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

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2

Compound Symbols:   c5_1, c6_1, c8_2, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2, c4_2, c4_1


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FILTERHIGH(z0, cons(z1, z2)) -> c8(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) by 
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2), GE(z0, 0))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2), GE(0, s(z0)))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2), GE(z0, 0))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2), GE(0, s(z0)))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2), GE(z0, 0))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2), GE(0, s(z0)))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2, c4_2, c4_1, c8_2


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

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_2, c2_2, c4_2, c4_1, c8_2, c8_1


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(z0, cons(z0, z1))), FILTERLOW(z0, cons(z0, z1))) by 
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c2_2, c4_2, c4_1, c8_2, c8_1, c1_2, c1_1


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

(73) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
We considered the (Usable) Rules:none
And the Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APPEND(x_1, x_2)) = 0
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(QSORT(x_1)) = [1]
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(false) = 0
   POL(filterhigh(x_1, x_2)) = x_1
   POL(filterlow(x_1, x_2)) = [1] + x_1
   POL(ge(x_1, x_2)) = 0
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = x_2
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(true) = 0
   POL(ys) = 0

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c2_2, c4_2, c4_1, c8_2, c8_1, c1_2, c1_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z0, z1)) -> c2(QSORT(filterhigh(z0, cons(z0, z1))), FILTERHIGH(z0, cons(z0, z1))) by 
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_2, c4_1, c8_2, c8_1, c1_2, c1_1, c2_2, c2_1


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

(77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
We considered the (Usable) Rules:none
And the Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APPEND(x_1, x_2)) = 0
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(QSORT(x_1)) = [1]
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(false) = 0
   POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2
   POL(filterlow(x_1, x_2)) = [1] + x_1
   POL(ge(x_1, x_2)) = 0
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(nil) = [1]
   POL(s(x_1)) = 0
   POL(true) = 0
   POL(ys) = 0

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_2, c4_1, c8_2, c8_1, c1_2, c1_1, c2_2, c2_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FILTERLOW(s(z0), cons(s(z1), x2)) -> c4(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) by 
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_2, c8_1, c1_2, c1_1, c2_2, c2_1, c4_2


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FILTERHIGH(s(z1), cons(s(z0), x2)) -> c8(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) by 
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_2, c1_1, c2_2, c2_1, c4_2, c8_2


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z0, z2)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, z2)), FILTERLOW(z0, cons(z0, z2))) by 
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_2, c2_1, c4_2, c8_2, c1_2


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

(85) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_2, c2_1, c4_2, c8_2, c1_2


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

(87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterhigh(z0, nil) -> nil
   filterlow(z0, nil) -> nil
   ge(0, s(z0)) -> false
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   ge(z0, 0) -> true
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
And the Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(APPEND(x_1, x_2)) = x_1
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = 0
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(QSORT(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(filterhigh(x_1, x_2)) = x_2
   POL(filterlow(x_1, x_2)) = x_2
   POL(ge(x_1, x_2)) = [1]
   POL(if1(x_1, x_2, x_3, x_4)) = x_1 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4
   POL(nil) = [1]
   POL(s(x_1)) = [1]
   POL(true) = 0
   POL(ys) = 0

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_2, c2_1, c4_2, c8_2, c1_2


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z0, z2)) -> c2(QSORT(if2(ge(z0, z0), z0, z0, z2)), FILTERHIGH(z0, cons(z0, z2))) by 
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2


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

(91) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2


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

(93) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterhigh(z0, nil) -> nil
   filterlow(z0, nil) -> nil
   ge(0, s(z0)) -> false
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   ge(z0, 0) -> true
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
And the Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APPEND(x_1, x_2)) = x_1
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(QSORT(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(false) = [1]
   POL(filterhigh(x_1, x_2)) = x_2
   POL(filterlow(x_1, x_2)) = x_1 + x_2
   POL(ge(x_1, x_2)) = [1]
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = x_1 + x_4
   POL(nil) = [1]
   POL(s(x_1)) = 0
   POL(true) = 0
   POL(ys) = 0

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2


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

(95) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1))) by QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2


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

(97) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1))) by QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
S tuples:
   IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2


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

(99) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(true, z0, z1, z2) -> c5(FILTERLOW(z0, z2)) by 
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
S tuples:
   IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1

Compound Symbols:   c6_1, c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1


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

(101) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(false, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) by 
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
S tuples:
   IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4

Compound Symbols:   c9_1, c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1


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

(103) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF2(true, z0, z1, z2) -> c9(FILTERHIGH(z0, z2)) by 
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
S tuples:
   IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF2_4, GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4

Compound Symbols:   c10_1, c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1


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

(105) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF2(false, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) by 
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
S tuples:
   GE(s(z0), s(z1)) -> c13(GE(z0, z1))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   GE_2, APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4

Compound Symbols:   c13_1, c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1


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

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(z0), s(z1)) -> c13(GE(z0, z1)) by 
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
S tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2

Compound Symbols:   c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1


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

(109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
S tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
K tuples:
   APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2

Compound Symbols:   c15_1, c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPEND(cons(z0, z1), ys) -> c15(APPEND(z1, ys)) by 
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
S tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
K tuples:
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1


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

(113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) by 
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
S tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
K tuples:
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1


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

(115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) by 
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
S tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1


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

(117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERLOW(s(x0), cons(s(x1), x2)) -> c4(GE(s(x0), s(x1))) by 
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
S tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0)))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1


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

(119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERHIGH(s(x0), cons(s(x1), x2)) -> c8(GE(s(x1), s(x0))) by 
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1)))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
S tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c1_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1


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

(121) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace QSORT(cons(x0, x1)) -> c1(FILTERLOW(x0, cons(x0, x1))) by 
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1)))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
S tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c2_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1


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

(123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace QSORT(cons(x0, x1)) -> c2(FILTERHIGH(x0, cons(x0, x1))) by 
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
S tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1


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

(125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(true, x0, 0, x1) -> c5(FILTERLOW(x0, x1)) by 
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
S tuples:
   FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1


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

(127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERLOW(z0, cons(0, x2)) -> c4(IF1(true, z0, 0, x2)) by 
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
S tuples:
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1


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

(129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(true, s(x0), s(0), x1) -> c5(FILTERLOW(s(x0), x1)) by 
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
S tuples:
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c5_1, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1


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

(131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(true, s(s(x0)), s(s(x1)), x2) -> c5(FILTERLOW(s(s(x0)), x2)) by 
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
S tuples:
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1, c5_1


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

(133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, 0, s(x0), x1) -> c6(FILTERLOW(0, x1)) by 
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
S tuples:
   FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c4_1, c8_1, c4_2, c8_2, c1_2, c2_2, c6_1, c9_1, c10_1, c13_1, c15_1, c1_1, c2_1, c5_1


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

(135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERLOW(0, cons(s(z0), x2)) -> c4(IF1(false, 0, s(z0), x2)) by 
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
S tuples:
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c8_1, c4_2, c8_2, c1_2, c2_2, c6_1, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1


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

(137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(0), s(s(x0)), x1) -> c6(FILTERLOW(s(0), x1)) by 
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
S tuples:
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c8_1, c4_2, c8_2, c1_2, c2_2, c6_1, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1


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

(139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2)) by 
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
S tuples:
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4

Compound Symbols:   c8_1, c4_2, c8_2, c1_2, c2_2, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1


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

(141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF2(true, 0, x0, x1) -> c9(FILTERHIGH(0, x1)) by 
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
S tuples:
   FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4

Compound Symbols:   c8_1, c4_2, c8_2, c1_2, c2_2, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1


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

(143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERHIGH(0, cons(z0, x2)) -> c8(IF2(true, 0, z0, x2)) by 
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
S tuples:
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4

Compound Symbols:   c8_1, c4_2, c8_2, c1_2, c2_2, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1


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

(145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF2(true, s(0), s(x0), x1) -> c9(FILTERHIGH(s(0), x1)) by 
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
S tuples:
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4

Compound Symbols:   c8_1, c4_2, c8_2, c1_2, c2_2, c9_1, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1


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

(147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF2(true, s(s(x0)), s(s(x1)), x2) -> c9(FILTERHIGH(s(s(x0)), x2)) by 
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
S tuples:
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4

Compound Symbols:   c8_1, c4_2, c8_2, c1_2, c2_2, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1


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

(149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF2(false, s(x0), 0, x1) -> c10(FILTERHIGH(s(x0), x1)) by 
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
S tuples:
   FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERHIGH_2, FILTERLOW_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4

Compound Symbols:   c8_1, c4_2, c8_2, c1_2, c2_2, c10_1, c13_1, c4_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1


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

(151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERHIGH(s(z0), cons(0, x2)) -> c8(IF2(false, s(z0), 0, x2)) by 
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
S tuples:
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4

Compound Symbols:   c4_2, c8_2, c1_2, c2_2, c10_1, c13_1, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1


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

(153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF2(false, s(s(x0)), s(0), x1) -> c10(FILTERHIGH(s(s(x0)), x1)) by 
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
S tuples:
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, IF2_4, GE_2, APPEND_2, IF1_4

Compound Symbols:   c4_2, c8_2, c1_2, c2_2, c10_1, c13_1, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1


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

(155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2)) by 
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
S tuples:
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, GE_2, APPEND_2, IF1_4, IF2_4

Compound Symbols:   c4_2, c8_2, c1_2, c2_2, c13_1, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1, c10_1


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

(157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(s(y0)), s(s(y1))) -> c13(GE(s(y0), s(y1))) by 
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
S tuples:
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, FILTERHIGH_2, QSORT_1, APPEND_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c4_2, c8_2, c1_2, c2_2, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1, c10_1, c13_1


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

(159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) by 
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
S tuples:
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERHIGH_2, QSORT_1, FILTERLOW_2, APPEND_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c8_2, c1_2, c2_2, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2


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

(161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c8(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1)))) by 
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
S tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   QSORT_1, FILTERLOW_2, FILTERHIGH_2, APPEND_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c1_2, c2_2, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2


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

(163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERLOW(s(z0), cons(s(0), x2)) -> c4(IF1(true, s(z0), s(0), x2)) by 
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
S tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   QSORT_1, FILTERLOW_2, FILTERHIGH_2, APPEND_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c1_2, c2_2, c4_1, c8_1, c15_1, c1_1, c2_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2


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

(165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c4(IF1(false, s(0), s(s(z0)), x2)) by 
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))
S tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   QSORT_1, FILTERHIGH_2, APPEND_2, FILTERLOW_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c1_2, c2_2, c8_1, c15_1, c1_1, c2_1, c4_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2


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

(167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERHIGH(s(0), cons(s(z0), x2)) -> c8(IF2(true, s(0), s(z0), x2)) by 
   FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1)))
   FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2)))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1)))
   FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2)))
S tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1)))
   FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2)))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   QSORT_1, FILTERHIGH_2, APPEND_2, FILTERLOW_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c1_2, c2_2, c8_1, c15_1, c1_1, c2_1, c4_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2


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

(169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c8(IF2(false, s(s(z0)), s(0), x2)) by 
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1)))
   FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))))
S tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1)))
   FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   QSORT_1, APPEND_2, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c1_2, c2_2, c15_1, c1_1, c2_1, c4_1, c8_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2


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

(171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPEND(cons(z0, cons(y0, y1)), ys) -> c15(APPEND(cons(y0, y1), ys)) by 
   APPEND(cons(z0, cons(z1, cons(y1, y2))), ys) -> c15(APPEND(cons(z1, cons(y1, y2)), ys))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1)))
   FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))))
   APPEND(cons(z0, cons(z1, cons(y1, y2))), ys) -> c15(APPEND(cons(z1, cons(y1, y2)), ys))
S tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1)))
   FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   APPEND(cons(z0, cons(z1, cons(y1, y2))), ys) -> c15(APPEND(cons(z1, cons(y1, y2)), ys))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c1_2, c2_2, c1_1, c2_1, c4_1, c8_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2, c15_1


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

(173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1))) by 
   QSORT(cons(0, cons(0, y1))) -> c1(FILTERLOW(0, cons(0, cons(0, y1))))
   QSORT(cons(0, cons(s(y0), y1))) -> c1(FILTERLOW(0, cons(0, cons(s(y0), y1))))

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(0, z0)), FILTERLOW(0, cons(0, z0)))
   QSORT(cons(0, z0)) -> c2(QSORT(filterhigh(0, z0)), FILTERHIGH(0, cons(0, z0)))
   QSORT(cons(0, z1)) -> c1(FILTERLOW(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1)))
   FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))))
   APPEND(cons(z0, cons(z1, cons(y1, y2))), ys) -> c15(APPEND(cons(z1, cons(y1, y2)), ys))
   QSORT(cons(0, cons(0, y1))) -> c1(FILTERLOW(0, cons(0, cons(0, y1))))
   QSORT(cons(0, cons(s(y0), y1))) -> c1(FILTERLOW(0, cons(0, cons(s(y0), y1))))
S tuples:
   QSORT(cons(s(z0), x1)) -> c1(QSORT(if1(ge(z0, z0), s(z0), s(z0), x1)), FILTERLOW(s(z0), cons(s(z0), x1)))
   QSORT(cons(s(z0), x1)) -> c2(QSORT(if2(ge(z0, z0), s(z0), s(z0), x1)), FILTERHIGH(s(z0), cons(s(z0), x1)))
   FILTERLOW(s(s(y0)), cons(s(s(y1)), z2)) -> c4(GE(s(s(y0)), s(s(y1))))
   FILTERHIGH(s(s(y1)), cons(s(s(y0)), z2)) -> c8(GE(s(s(y0)), s(s(y1))))
   IF1(true, z0, 0, cons(0, y1)) -> c5(FILTERLOW(z0, cons(0, y1)))
   IF1(true, 0, 0, cons(s(y0), y1)) -> c5(FILTERLOW(0, cons(s(y0), y1)))
   IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(y0), 0, cons(s(0), y1)) -> c5(FILTERLOW(s(y0), cons(s(0), y1)))
   IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   FILTERLOW(z0, cons(0, cons(0, y1))) -> c4(IF1(true, z0, 0, cons(0, y1)))
   FILTERLOW(0, cons(0, cons(s(y0), y1))) -> c4(IF1(true, 0, 0, cons(s(y0), y1)))
   FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERLOW(s(y0), cons(0, cons(s(0), y1))) -> c4(IF1(true, s(y0), 0, cons(s(0), y1)))
   FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), 0, cons(s(s(y0)), y1)))
   IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(y0)), cons(s(s(y1)), y2)))
   IF1(true, s(z0), s(0), cons(s(0), y1)) -> c5(FILTERLOW(s(z0), cons(s(0), y1)))
   IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c5(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(0, y1))))
   IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(y0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(z0), cons(0, cons(s(0), y1))))
   IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c5(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c5(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c5(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c5(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF1(false, 0, s(z0), cons(s(y0), y1)) -> c6(FILTERLOW(0, cons(s(y0), y1)))
   IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c6(FILTERLOW(0, cons(0, cons(0, y1))))
   IF1(false, 0, s(z0), cons(0, cons(s(y0), y1))) -> c6(FILTERLOW(0, cons(0, cons(s(y0), y1))))
   FILTERLOW(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF1(false, 0, s(z0), cons(s(y1), y2)))
   FILTERLOW(0, cons(s(z0), cons(0, cons(0, y1)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(0, y1))))
   FILTERLOW(0, cons(s(z0), cons(0, cons(s(y1), y2)))) -> c4(IF1(false, 0, s(z0), cons(0, cons(s(y1), y2))))
   IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c6(FILTERLOW(s(0), cons(s(0), y1)))
   IF1(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(FILTERLOW(s(0), cons(s(s(y0)), y1)))
   IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(0), cons(0, cons(0, y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(0), y1))))
   IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(FILTERLOW(s(0), cons(0, cons(s(s(y0)), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(FILTERLOW(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(FILTERLOW(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(0, y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(FILTERLOW(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(true, 0, z0, cons(y0, y1)) -> c9(FILTERHIGH(0, cons(y0, y1)))
   FILTERHIGH(0, cons(z0, cons(y1, y2))) -> c8(IF2(true, 0, z0, cons(y1, y2)))
   IF2(true, s(0), s(z0), cons(0, y1)) -> c9(FILTERHIGH(s(0), cons(0, y1)))
   IF2(true, s(0), s(z0), cons(s(y0), y1)) -> c9(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c9(FILTERHIGH(s(s(z0)), cons(0, y1)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c9(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c9(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(z0), 0, cons(0, y1)) -> c10(FILTERHIGH(s(z0), cons(0, y1)))
   IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(0), 0, cons(s(y0), y1)) -> c10(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c10(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   FILTERHIGH(s(z0), cons(0, cons(0, y1))) -> c8(IF2(false, s(z0), 0, cons(0, y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)))
   FILTERHIGH(s(0), cons(0, cons(s(y0), y1))) -> c8(IF2(false, s(0), 0, cons(s(y0), y1)))
   FILTERHIGH(s(s(y0)), cons(0, cons(s(0), y1))) -> c8(IF2(false, s(s(y0)), 0, cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(0), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c10(FILTERHIGH(s(s(z0)), cons(s(s(y1)), y2)))
   IF2(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c10(FILTERHIGH(s(s(z0)), cons(s(0), y1)))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(0, y1))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   IF2(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c10(FILTERHIGH(s(s(z0)), cons(0, cons(s(0), y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c13(GE(s(s(y0)), s(s(y1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF1(ge(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(s(0), y3)), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c8(IF2(ge(z1, z0), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), GE(s(s(z1)), s(s(z0))))
   FILTERHIGH(s(s(s(y1))), cons(s(s(s(y0))), z2)) -> c8(IF2(ge(s(y0), s(y1)), s(s(s(y1))), s(s(s(y0))), z2), GE(s(s(s(y0))), s(s(s(y1)))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)))
   FILTERLOW(s(z0), cons(s(0), cons(s(0), y1))) -> c4(IF1(true, s(z0), s(0), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(0), cons(s(s(y0)), y1))) -> c4(IF1(true, s(0), s(0), cons(s(s(y0)), y1)))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(0, y1))))
   FILTERLOW(s(s(y0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERLOW(s(z0), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c4(IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(0), y1)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))))
   FILTERLOW(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c4(IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(0), cons(s(z0), cons(0, y1))) -> c8(IF2(true, s(0), s(z0), cons(0, y1)))
   FILTERHIGH(s(0), cons(s(z0), cons(s(y1), y2))) -> c8(IF2(true, s(0), s(z0), cons(s(y1), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(s(y1)), y2)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c8(IF2(false, s(s(z0)), s(0), cons(s(0), y1)))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(0, y1))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))))
   FILTERHIGH(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c8(IF2(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))))
K tuples:
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, 0, 0, x1)), FILTERLOW(0, cons(0, x1)))
   QSORT(cons(0, x1)) -> c2(QSORT(if2(true, 0, 0, x1)), FILTERHIGH(0, cons(0, x1)))
   QSORT(cons(s(s(y0)), z1)) -> c1(FILTERLOW(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c1(FILTERLOW(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c1(FILTERLOW(s(0), cons(s(0), z1)))
   QSORT(cons(0, z1)) -> c2(FILTERHIGH(0, cons(0, z1)))
   QSORT(cons(s(s(y0)), z1)) -> c2(FILTERHIGH(s(s(y0)), cons(s(s(y0)), z1)))
   QSORT(cons(s(y0), z1)) -> c2(FILTERHIGH(s(y0), cons(s(y0), z1)))
   QSORT(cons(s(0), z1)) -> c2(FILTERHIGH(s(0), cons(s(0), z1)))
   APPEND(cons(z0, cons(z1, cons(y1, y2))), ys) -> c15(APPEND(cons(z1, cons(y1, y2)), ys))
   QSORT(cons(0, cons(0, y1))) -> c1(FILTERLOW(0, cons(0, cons(0, y1))))
   QSORT(cons(0, cons(s(y0), y1))) -> c1(FILTERLOW(0, cons(0, cons(s(y0), y1))))
Defined Rule Symbols:   ge_2, filterlow_2, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2, APPEND_2

Compound Symbols:   c1_2, c2_2, c1_1, c2_1, c4_1, c8_1, c5_1, c6_1, c9_1, c10_1, c13_1, c4_2, c8_2, c15_1