KILLED
proof of input_1DQUF7bWzY.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 8 ms]
        (14) CpxRNTS
        (15) InliningProof [UPPER BOUND(ID), 668 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 350 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 140 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 182 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 465 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 145 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 1647 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 226 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 4093 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 932 ms]
        (50) CpxRNTS
        (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 1431 ms]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 187 ms]
        (56) CpxRNTS
    (57) CompletionProof [UPPER BOUND(ID), 0 ms]
    (58) CpxTypedWeightedCompleteTrs
        (59) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (60) CpxRNTS
(61) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(62) CdtProblem
    (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 153 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 65 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 386 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 315 ms]
    (90) CdtProblem
    (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtRewritingProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 2 ms]
    (168) CdtProblem
    (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) 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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   minsort(nil) -> nil
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
   min(x, nil) -> x
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))

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:

   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   eq(0', 0') -> true
   eq(0', s(y)) -> false
   eq(s(x), 0') -> false
   eq(s(x), s(y)) -> eq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   minsort(nil) -> nil
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
   min(x, nil) -> x
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))

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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   minsort(nil) -> nil
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
   min(x, nil) -> x
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))

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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   eq(0, 0) -> true [1]
   eq(0, s(y)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   minsort(nil) -> nil [1]
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1]
   min(x, nil) -> x [1]
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) [1]
   del(x, nil) -> nil [1]
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) [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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   eq(0, 0) -> true [1]
   eq(0, s(y)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   minsort(nil) -> nil [1]
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1]
   min(x, nil) -> x [1]
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) [1]
   del(x, nil) -> nil [1]
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) [1]

The TRS has the following type information:
le :: 0:s:nil:cons -> 0:s:nil:cons -> true:false
0 :: 0:s:nil:cons
true :: true:false
s :: 0:s:nil:cons -> 0:s:nil:cons
false :: true:false
eq :: 0:s:nil:cons -> 0:s:nil:cons -> true:false
if :: true:false -> 0:s:nil:cons -> 0:s:nil:cons -> 0:s:nil:cons
minsort :: 0:s:nil:cons -> 0:s:nil:cons
nil :: 0:s:nil:cons
cons :: 0:s:nil:cons -> 0:s:nil:cons -> 0:s:nil:cons
min :: 0:s:nil:cons -> 0:s:nil:cons -> 0:s:nil:cons
del :: 0:s:nil:cons -> 0:s:nil:cons -> 0:s:nil:cons

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

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

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

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   minsort_1

(c) The following functions are completely defined:

   le_2
   min_2
   del_2
   eq_2
   if_3

Due to the following rules being added:

   le(v0, v1) -> null_le [0]
   min(v0, v1) -> null_min [0]
   del(v0, v1) -> null_del [0]
   eq(v0, v1) -> null_eq [0]
   if(v0, v1, v2) -> null_if [0]

And the following fresh constants:    null_le, null_min, null_del, null_eq, null_if

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

(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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   eq(0, 0) -> true [1]
   eq(0, s(y)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   minsort(nil) -> nil [1]
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1]
   min(x, nil) -> x [1]
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) [1]
   del(x, nil) -> nil [1]
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) [1]
   le(v0, v1) -> null_le [0]
   min(v0, v1) -> null_min [0]
   del(v0, v1) -> null_del [0]
   eq(v0, v1) -> null_eq [0]
   if(v0, v1, v2) -> null_if [0]

The TRS has the following type information:
le :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> true:false:null_le:null_eq
0 :: 0:s:nil:cons:null_min:null_del:null_if
true :: true:false:null_le:null_eq
s :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
false :: true:false:null_le:null_eq
eq :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> true:false:null_le:null_eq
if :: true:false:null_le:null_eq -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
minsort :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
nil :: 0:s:nil:cons:null_min:null_del:null_if
cons :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
min :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
del :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
null_le :: true:false:null_le:null_eq
null_min :: 0:s:nil:cons:null_min:null_del:null_if
null_del :: 0:s:nil:cons:null_min:null_del:null_if
null_eq :: true:false:null_le:null_eq
null_if :: 0:s:nil:cons:null_min:null_del:null_if

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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   eq(0, 0) -> true [1]
   eq(0, s(y)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   minsort(nil) -> nil [1]
   minsort(cons(x, nil)) -> cons(min(x, nil), minsort(del(x, cons(x, nil)))) [2]
   minsort(cons(x, cons(y', z'))) -> cons(min(x, cons(y', z')), minsort(del(if(le(x, y'), min(x, z'), min(y', z')), cons(x, cons(y', z'))))) [2]
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(null_min, cons(x, y)))) [1]
   min(x, nil) -> x [1]
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) [1]
   del(x, nil) -> nil [1]
   del(0, cons(0, nil)) -> if(true, nil, cons(0, nil)) [3]
   del(0, cons(0, cons(y11, z8))) -> if(true, cons(y11, z8), cons(0, if(eq(0, y11), z8, cons(y11, del(0, z8))))) [3]
   del(0, cons(0, z)) -> if(true, z, cons(0, null_del)) [2]
   del(0, cons(s(y9), nil)) -> if(false, nil, cons(s(y9), nil)) [3]
   del(0, cons(s(y9), cons(y12, z9))) -> if(false, cons(y12, z9), cons(s(y9), if(eq(0, y12), z9, cons(y12, del(0, z9))))) [3]
   del(0, cons(s(y9), z)) -> if(false, z, cons(s(y9), null_del)) [2]
   del(s(x1), cons(0, nil)) -> if(false, nil, cons(0, nil)) [3]
   del(s(x1), cons(0, cons(y13, z10))) -> if(false, cons(y13, z10), cons(0, if(eq(s(x1), y13), z10, cons(y13, del(s(x1), z10))))) [3]
   del(s(x1), cons(0, z)) -> if(false, z, cons(0, null_del)) [2]
   del(s(x2), cons(s(y10), nil)) -> if(eq(x2, y10), nil, cons(s(y10), nil)) [3]
   del(s(x2), cons(s(y10), cons(y14, z11))) -> if(eq(x2, y10), cons(y14, z11), cons(s(y10), if(eq(s(x2), y14), z11, cons(y14, del(s(x2), z11))))) [3]
   del(s(x2), cons(s(y10), z)) -> if(eq(x2, y10), z, cons(s(y10), null_del)) [2]
   del(x, cons(y, nil)) -> if(null_eq, nil, cons(y, nil)) [2]
   del(x, cons(y, cons(y15, z12))) -> if(null_eq, cons(y15, z12), cons(y, if(eq(x, y15), z12, cons(y15, del(x, z12))))) [2]
   del(x, cons(y, z)) -> if(null_eq, z, cons(y, null_del)) [1]
   le(v0, v1) -> null_le [0]
   min(v0, v1) -> null_min [0]
   del(v0, v1) -> null_del [0]
   eq(v0, v1) -> null_eq [0]
   if(v0, v1, v2) -> null_if [0]

The TRS has the following type information:
le :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> true:false:null_le:null_eq
0 :: 0:s:nil:cons:null_min:null_del:null_if
true :: true:false:null_le:null_eq
s :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
false :: true:false:null_le:null_eq
eq :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> true:false:null_le:null_eq
if :: true:false:null_le:null_eq -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
minsort :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
nil :: 0:s:nil:cons:null_min:null_del:null_if
cons :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
min :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
del :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if
null_le :: true:false:null_le:null_eq
null_min :: 0:s:nil:cons:null_min:null_del:null_if
null_del :: 0:s:nil:cons:null_min:null_del:null_if
null_eq :: true:false:null_le:null_eq
null_if :: 0:s:nil:cons:null_min:null_del:null_if

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

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

   0 => 0
   true => 2
   false => 1
   nil => 1
   null_le => 0
   null_min => 0
   null_del => 0
   null_eq => 0
   null_if => 0

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

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

   del(z'', z1) -{ 2 }-> if(eq(x2, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' = 1 + x2, x2 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(x2, y10), 1, 1 + (1 + y10) + 1) :|: y10 >= 0, z'' = 1 + x2, z1 = 1 + (1 + y10) + 1, x2 >= 0
   del(z'', z1) -{ 3 }-> if(eq(x2, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + x2, y14), z11, 1 + y14 + del(1 + x2, z11))) :|: z11 >= 0, y10 >= 0, z'' = 1 + x2, z1 = 1 + (1 + y10) + (1 + y14 + z11), x2 >= 0, y14 >= 0
   del(z'', z1) -{ 2 }-> if(2, z, 1 + 0 + 0) :|: z'' = 0, z >= 0, z1 = 1 + 0 + z
   del(z'', z1) -{ 3 }-> if(2, 1, 1 + 0 + 1) :|: z'' = 0, z1 = 1 + 0 + 1
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 2 }-> if(1, z, 1 + 0 + 0) :|: x1 >= 0, z >= 0, z'' = 1 + x1, z1 = 1 + 0 + z
   del(z'', z1) -{ 2 }-> if(1, z, 1 + (1 + y9) + 0) :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z
   del(z'', z1) -{ 3 }-> if(1, 1, 1 + 0 + 1) :|: x1 >= 0, z'' = 1 + x1, z1 = 1 + 0 + 1
   del(z'', z1) -{ 3 }-> if(1, 1, 1 + (1 + y9) + 1) :|: z'' = 0, y9 >= 0, z1 = 1 + (1 + y9) + 1
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + x1, y13), z10, 1 + y13 + del(1 + x1, z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), x1 >= 0, z'' = 1 + x1, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 1 }-> if(0, z, 1 + y + 0) :|: z >= 0, z1 = 1 + y + z, x >= 0, y >= 0, z'' = x
   del(z'', z1) -{ 2 }-> if(0, 1, 1 + y + 1) :|: x >= 0, y >= 0, z'' = x, z1 = 1 + y + 1
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(x, y15), z12, 1 + y15 + del(x, z12))) :|: z1 = 1 + y + (1 + y15 + z12), x >= 0, y >= 0, z'' = x, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: x >= 0, z1 = 1, z'' = x
   del(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   eq(z'', z1) -{ 1 }-> eq(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, y >= 0, z1 = 1 + y
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, x >= 0, z'' = 1 + x
   eq(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   if(z'', z1, z2) -{ 1 }-> x :|: z2 = y, x >= 0, y >= 0, z'' = 2, z1 = x
   if(z'', z1, z2) -{ 1 }-> y :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' = v0, z2 = v2, v0 >= 0, z1 = v1, v1 >= 0, v2 >= 0
   le(z'', z1) -{ 1 }-> le(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = y, y >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, x >= 0, z'' = 1 + x
   le(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   min(z'', z1) -{ 1 }-> x :|: x >= 0, z1 = 1, z'' = x
   min(z'', z1) -{ 1 }-> if(le(x, y), min(x, z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, x >= 0, y >= 0, z'' = x
   min(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1) + minsort(del(x, 1 + x + 1)) :|: x >= 0, z'' = 1 + x + 1
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0


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

(15) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   if(z'', z1, z2) -{ 1 }-> x :|: z2 = y, x >= 0, y >= 0, z'' = 2, z1 = x
   if(z'', z1, z2) -{ 1 }-> y :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' = v0, z2 = v2, v0 >= 0, z1 = v1, v1 >= 0, v2 >= 0

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z >= 0, z1 = 1 + 0 + z, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, y9 >= 0, z1 = 1 + (1 + y9) + 1, 1 + (1 + y9) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: x1 >= 0, z'' = 1 + x1, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: x1 >= 0, z >= 0, z'' = 1 + x1, z1 = 1 + 0 + z, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 2 }-> if(eq(x2, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' = 1 + x2, x2 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(x2, y10), 1, 1 + (1 + y10) + 1) :|: y10 >= 0, z'' = 1 + x2, z1 = 1 + (1 + y10) + 1, x2 >= 0
   del(z'', z1) -{ 3 }-> if(eq(x2, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + x2, y14), z11, 1 + y14 + del(1 + x2, z11))) :|: z11 >= 0, y10 >= 0, z'' = 1 + x2, z1 = 1 + (1 + y10) + (1 + y14 + z11), x2 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + x1, y13), z10, 1 + y13 + del(1 + x1, z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), x1 >= 0, z'' = 1 + x1, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(x, y15), z12, 1 + y15 + del(x, z12))) :|: z1 = 1 + y + (1 + y15 + z12), x >= 0, y >= 0, z'' = x, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: x >= 0, z1 = 1, z'' = x
   del(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + 0 + z, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, y9 >= 0, z1 = 1 + (1 + y9) + 1, 1 = v0, 1 + (1 + y9) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: x1 >= 0, z'' = 1 + x1, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: x1 >= 0, z >= 0, z'' = 1 + x1, z1 = 1 + 0 + z, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: x >= 0, y >= 0, z'' = x, z1 = 1 + y + 1, 0 = v0, 1 + y + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, x >= 0, y >= 0, z'' = x, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, y >= 0, z1 = 1 + y
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, x >= 0, z'' = 1 + x
   eq(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   if(z'', z1, z2) -{ 1 }-> x :|: z2 = y, x >= 0, y >= 0, z'' = 2, z1 = x
   if(z'', z1, z2) -{ 1 }-> y :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' = v0, z2 = v2, v0 >= 0, z1 = v1, v1 >= 0, v2 >= 0
   le(z'', z1) -{ 1 }-> le(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = y, y >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, x >= 0, z'' = 1 + x
   le(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   min(z'', z1) -{ 1 }-> x :|: x >= 0, z1 = 1, z'' = x
   min(z'', z1) -{ 1 }-> if(le(x, y), min(x, z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, x >= 0, y >= 0, z'' = x
   min(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1) + minsort(del(x, 1 + x + 1)) :|: x >= 0, z'' = 1 + x + 1
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0


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

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 1 }-> if(le(z'', y), min(z'', z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0


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

   { le }
   { if }
   { eq }
   { min }
   { del }
   { minsort }

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 1 }-> if(le(z'', y), min(z'', z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {le}, {if}, {eq}, {min}, {del}, {minsort}

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 1 }-> if(le(z'', y), min(z'', z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {le}, {if}, {eq}, {min}, {del}, {minsort}

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(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 1 }-> if(le(z'', y), min(z'', z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {le}, {if}, {eq}, {min}, {del}, {minsort}
Previous analysis results are:
  le: runtime: ?, size: O(1) [2]

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(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z1

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 1 }-> if(le(z'', y), min(z'', z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {if}, {eq}, {min}, {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]

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

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {if}, {eq}, {min}, {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z1 + z2

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {if}, {eq}, {min}, {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: ?, size: O(n^1) [z1 + z2]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {eq}, {min}, {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]

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

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {eq}, {min}, {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {eq}, {min}, {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: ?, size: O(1) [2]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z''

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

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

   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {min}, {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: O(n^1) [3 + z''], size: O(1) [2]

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

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

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

   del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {min}, {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: O(n^1) [3 + z''], size: O(1) [2]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {min}, {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: O(n^1) [3 + z''], size: O(1) [2]
  min: runtime: ?, size: EXP

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0

Function symbols to be analyzed: {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: O(n^1) [3 + z''], size: O(1) [2]
  min: runtime: EXP, size: EXP

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

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

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

   del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 2 + inf }-> 1 + s12 + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: s12 >= 0, s12 <= inf', z'' - 2 >= 0
   minsort(z'') -{ 1 + inf'' }-> 1 + s13 + minsort(del(0, 1 + x + y)) :|: s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 5 + inf2 + inf4 + inf6 + y' }-> 1 + s14 + minsort(del(s17, 1 + x + (1 + y' + z'))) :|: s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0

Function symbols to be analyzed: {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: O(n^1) [3 + z''], size: O(1) [2]
  min: runtime: EXP, size: EXP

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: del
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 11*z1 + 4*z1^2

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

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

   del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 2 + inf }-> 1 + s12 + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: s12 >= 0, s12 <= inf', z'' - 2 >= 0
   minsort(z'') -{ 1 + inf'' }-> 1 + s13 + minsort(del(0, 1 + x + y)) :|: s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 5 + inf2 + inf4 + inf6 + y' }-> 1 + s14 + minsort(del(s17, 1 + x + (1 + y' + z'))) :|: s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0

Function symbols to be analyzed: {del}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: O(n^1) [3 + z''], size: O(1) [2]
  min: runtime: EXP, size: EXP
  del: runtime: ?, size: O(n^2) [8 + 11*z1 + 4*z1^2]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: del
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 45 + 7*z'' + 2*z''*z1 + 13*z1

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 2 + inf }-> 1 + s12 + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: s12 >= 0, s12 <= inf', z'' - 2 >= 0
   minsort(z'') -{ 1 + inf'' }-> 1 + s13 + minsort(del(0, 1 + x + y)) :|: s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 5 + inf2 + inf4 + inf6 + y' }-> 1 + s14 + minsort(del(s17, 1 + x + (1 + y' + z'))) :|: s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0

Function symbols to be analyzed: {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: O(n^1) [3 + z''], size: O(1) [2]
  min: runtime: EXP, size: EXP
  del: runtime: O(n^2) [45 + 7*z'' + 2*z''*z1 + 13*z1], size: O(n^2) [8 + 11*z1 + 4*z1^2]

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

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

   del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 53 + 13*z8 }-> s26 :|: s24 >= 0, s24 <= 8 + 11 * z8 + 4 * (z8 * z8), s25 >= 0, s25 <= 1 + y11 + s24 + z8, s26 >= 0, s26 <= 1 + 0 + s25 + (1 + y11 + z8), s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 53 + 13*z9 }-> s29 :|: s27 >= 0, s27 <= 8 + 11 * z9 + 4 * (z9 * z9), s28 >= 0, s28 <= 1 + y12 + s27 + z9, s29 >= 0, s29 <= 1 + (1 + y9) + s28 + (1 + y12 + z9), s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 53 + 8*z'' + 2*z''*z10 + 13*z10 }-> s32 :|: s30 >= 0, s30 <= 8 + 11 * z10 + 4 * (z10 * z10), s31 >= 0, s31 <= 1 + y13 + s30 + z10, s32 >= 0, s32 <= 1 + 0 + s31 + (1 + y13 + z10), s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 55 + 9*z'' + 2*z''*z11 + 13*z11 }-> s35 :|: s33 >= 0, s33 <= 8 + 11 * z11 + 4 * (z11 * z11), s34 >= 0, s34 <= 1 + y14 + s33 + z11, s35 >= 0, s35 <= 1 + (1 + y10) + s34 + (1 + y14 + z11), s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 52 + 8*z'' + 2*z''*z12 + 13*z12 }-> s38 :|: s36 >= 0, s36 <= 8 + 11 * z12 + 4 * (z12 * z12), s37 >= 0, s37 <= 1 + y15 + s36 + z12, s38 >= 0, s38 <= 1 + y + s37 + (1 + y15 + z12), s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 33 + inf + 16*z'' + 2*z''^2 }-> 1 + s12 + minsort(s21) :|: s21 >= 0, s21 <= 8 + 11 * (1 + (z'' - 2) + 1) + 4 * ((1 + (z'' - 2) + 1) * (1 + (z'' - 2) + 1)), s12 >= 0, s12 <= inf', z'' - 2 >= 0
   minsort(z'') -{ 59 + inf'' + 13*x + 13*y }-> 1 + s13 + minsort(s22) :|: s22 >= 0, s22 <= 8 + 11 * (1 + x + y) + 4 * ((1 + x + y) * (1 + x + y)), s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 76 + inf2 + inf4 + inf6 + 11*s17 + 2*s17*x + 2*s17*y' + 2*s17*z' + 13*x + 14*y' + 13*z' }-> 1 + s14 + minsort(s23) :|: s23 >= 0, s23 <= 8 + 11 * (1 + x + (1 + y' + z')) + 4 * ((1 + x + (1 + y' + z')) * (1 + x + (1 + y' + z'))), s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0

Function symbols to be analyzed: {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: O(n^1) [3 + z''], size: O(1) [2]
  min: runtime: EXP, size: EXP
  del: runtime: O(n^2) [45 + 7*z'' + 2*z''*z1 + 13*z1], size: O(n^2) [8 + 11*z1 + 4*z1^2]

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(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 53 + 13*z8 }-> s26 :|: s24 >= 0, s24 <= 8 + 11 * z8 + 4 * (z8 * z8), s25 >= 0, s25 <= 1 + y11 + s24 + z8, s26 >= 0, s26 <= 1 + 0 + s25 + (1 + y11 + z8), s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 53 + 13*z9 }-> s29 :|: s27 >= 0, s27 <= 8 + 11 * z9 + 4 * (z9 * z9), s28 >= 0, s28 <= 1 + y12 + s27 + z9, s29 >= 0, s29 <= 1 + (1 + y9) + s28 + (1 + y12 + z9), s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 53 + 8*z'' + 2*z''*z10 + 13*z10 }-> s32 :|: s30 >= 0, s30 <= 8 + 11 * z10 + 4 * (z10 * z10), s31 >= 0, s31 <= 1 + y13 + s30 + z10, s32 >= 0, s32 <= 1 + 0 + s31 + (1 + y13 + z10), s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 55 + 9*z'' + 2*z''*z11 + 13*z11 }-> s35 :|: s33 >= 0, s33 <= 8 + 11 * z11 + 4 * (z11 * z11), s34 >= 0, s34 <= 1 + y14 + s33 + z11, s35 >= 0, s35 <= 1 + (1 + y10) + s34 + (1 + y14 + z11), s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 52 + 8*z'' + 2*z''*z12 + 13*z12 }-> s38 :|: s36 >= 0, s36 <= 8 + 11 * z12 + 4 * (z12 * z12), s37 >= 0, s37 <= 1 + y15 + s36 + z12, s38 >= 0, s38 <= 1 + y + s37 + (1 + y15 + z12), s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 33 + inf + 16*z'' + 2*z''^2 }-> 1 + s12 + minsort(s21) :|: s21 >= 0, s21 <= 8 + 11 * (1 + (z'' - 2) + 1) + 4 * ((1 + (z'' - 2) + 1) * (1 + (z'' - 2) + 1)), s12 >= 0, s12 <= inf', z'' - 2 >= 0
   minsort(z'') -{ 59 + inf'' + 13*x + 13*y }-> 1 + s13 + minsort(s22) :|: s22 >= 0, s22 <= 8 + 11 * (1 + x + y) + 4 * ((1 + x + y) * (1 + x + y)), s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 76 + inf2 + inf4 + inf6 + 11*s17 + 2*s17*x + 2*s17*y' + 2*s17*z' + 13*x + 14*y' + 13*z' }-> 1 + s14 + minsort(s23) :|: s23 >= 0, s23 <= 8 + 11 * (1 + x + (1 + y' + z')) + 4 * ((1 + x + (1 + y' + z')) * (1 + x + (1 + y' + z'))), s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0

Function symbols to be analyzed: {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: O(n^1) [3 + z''], size: O(1) [2]
  min: runtime: EXP, size: EXP
  del: runtime: O(n^2) [45 + 7*z'' + 2*z''*z1 + 13*z1], size: O(n^2) [8 + 11*z1 + 4*z1^2]
  minsort: runtime: ?, size: INF

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z
   del(z'', z1) -{ 53 + 13*z8 }-> s26 :|: s24 >= 0, s24 <= 8 + 11 * z8 + 4 * (z8 * z8), s25 >= 0, s25 <= 1 + y11 + s24 + z8, s26 >= 0, s26 <= 1 + 0 + s25 + (1 + y11 + z8), s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0
   del(z'', z1) -{ 53 + 13*z9 }-> s29 :|: s27 >= 0, s27 <= 8 + 11 * z9 + 4 * (z9 * z9), s28 >= 0, s28 <= 1 + y12 + s27 + z9, s29 >= 0, s29 <= 1 + (1 + y9) + s28 + (1 + y12 + z9), s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9)
   del(z'', z1) -{ 53 + 8*z'' + 2*z''*z10 + 13*z10 }-> s32 :|: s30 >= 0, s30 <= 8 + 11 * z10 + 4 * (z10 * z10), s31 >= 0, s31 <= 1 + y13 + s30 + z10, s32 >= 0, s32 <= 1 + 0 + s31 + (1 + y13 + z10), s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0
   del(z'', z1) -{ 55 + 9*z'' + 2*z''*z11 + 13*z11 }-> s35 :|: s33 >= 0, s33 <= 8 + 11 * z11 + 4 * (z11 * z11), s34 >= 0, s34 <= 1 + y14 + s33 + z11, s35 >= 0, s35 <= 1 + (1 + y10) + s34 + (1 + y14 + z11), s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0
   del(z'', z1) -{ 52 + 8*z'' + 2*z''*z12 + 13*z12 }-> s38 :|: s36 >= 0, s36 <= 8 + 11 * z12 + 4 * (z12 * z12), s37 >= 0, s37 <= 1 + y15 + s36 + z12, s38 >= 0, s38 <= 1 + y + s37 + (1 + y15 + z12), s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0
   del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x
   del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x
   del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x
   del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x
   del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x
   del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1
   del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0
   eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2
   if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1
   if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0
   le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1
   min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   minsort(z'') -{ 1 }-> 1 :|: z'' = 1
   minsort(z'') -{ 33 + inf + 16*z'' + 2*z''^2 }-> 1 + s12 + minsort(s21) :|: s21 >= 0, s21 <= 8 + 11 * (1 + (z'' - 2) + 1) + 4 * ((1 + (z'' - 2) + 1) * (1 + (z'' - 2) + 1)), s12 >= 0, s12 <= inf', z'' - 2 >= 0
   minsort(z'') -{ 59 + inf'' + 13*x + 13*y }-> 1 + s13 + minsort(s22) :|: s22 >= 0, s22 <= 8 + 11 * (1 + x + y) + 4 * ((1 + x + y) * (1 + x + y)), s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0
   minsort(z'') -{ 76 + inf2 + inf4 + inf6 + 11*s17 + 2*s17*x + 2*s17*y' + 2*s17*z' + 13*x + 14*y' + 13*z' }-> 1 + s14 + minsort(s23) :|: s23 >= 0, s23 <= 8 + 11 * (1 + x + (1 + y' + z')) + 4 * ((1 + x + (1 + y' + z')) * (1 + x + (1 + y' + z'))), s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0

Function symbols to be analyzed: {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [2]
  if: runtime: O(1) [1], size: O(n^1) [z1 + z2]
  eq: runtime: O(n^1) [3 + z''], size: O(1) [2]
  min: runtime: EXP, size: EXP
  del: runtime: O(n^2) [45 + 7*z'' + 2*z''*z1 + 13*z1], size: O(n^2) [8 + 11*z1 + 4*z1^2]
  minsort: runtime: INF, size: INF

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

(57) 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:

   le(v0, v1) -> null_le [0]
   eq(v0, v1) -> null_eq [0]
   minsort(v0) -> null_minsort [0]
   min(v0, v1) -> null_min [0]
   del(v0, v1) -> null_del [0]
   if(v0, v1, v2) -> null_if [0]

And the following fresh constants:    null_le, null_eq, null_minsort, null_min, null_del, null_if

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

(58)
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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   eq(0, 0) -> true [1]
   eq(0, s(y)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   minsort(nil) -> nil [1]
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1]
   min(x, nil) -> x [1]
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) [1]
   del(x, nil) -> nil [1]
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) [1]
   le(v0, v1) -> null_le [0]
   eq(v0, v1) -> null_eq [0]
   minsort(v0) -> null_minsort [0]
   min(v0, v1) -> null_min [0]
   del(v0, v1) -> null_del [0]
   if(v0, v1, v2) -> null_if [0]

The TRS has the following type information:
le :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> true:false:null_le:null_eq
0 :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if
true :: true:false:null_le:null_eq
s :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if
false :: true:false:null_le:null_eq
eq :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> true:false:null_le:null_eq
if :: true:false:null_le:null_eq -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if
minsort :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if
nil :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if
cons :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if
min :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if
del :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if
null_le :: true:false:null_le:null_eq
null_eq :: true:false:null_le:null_eq
null_minsort :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if
null_min :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if
null_del :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if
null_if :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if

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

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

   0 => 0
   true => 2
   false => 1
   nil => 1
   null_le => 0
   null_eq => 0
   null_minsort => 0
   null_min => 0
   null_del => 0
   null_if => 0

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   del(z', z'') -{ 1 }-> if(eq(x, y), z, 1 + y + del(x, z)) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
   del(z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1
   del(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   eq(z', z'') -{ 1 }-> eq(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   eq(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   if(z', z'', z1) -{ 1 }-> x :|: z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
   if(z', z'', z1) -{ 1 }-> y :|: z1 = y, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   min(z', z'') -{ 1 }-> x :|: z' = x, x >= 0, z'' = 1
   min(z', z'') -{ 1 }-> if(le(x, y), min(x, z), min(y, z)) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
   min(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   minsort(z') -{ 1 }-> 1 :|: z' = 1
   minsort(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   minsort(z') -{ 1 }-> 1 + min(x, y) + minsort(del(min(x, y), 1 + x + y)) :|: z' = 1 + x + y, x >= 0, y >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   minsort(nil) -> nil
   minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1))))
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   del(z0, nil) -> nil
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(0, 0) -> c3
   EQ(0, s(z0)) -> c4
   EQ(s(z0), 0) -> c5
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF(true, z0, z1) -> c7
   IF(false, z0, z1) -> c8
   MINSORT(nil) -> c9
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, nil) -> c12
   MIN(z0, cons(z1, z2)) -> c13(IF(le(z0, z1), min(z0, z2), min(z1, z2)), LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z1, z2))
   DEL(z0, nil) -> c16
   DEL(z0, cons(z1, z2)) -> c17(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), DEL(z0, z2))
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(0, 0) -> c3
   EQ(0, s(z0)) -> c4
   EQ(s(z0), 0) -> c5
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF(true, z0, z1) -> c7
   IF(false, z0, z1) -> c8
   MINSORT(nil) -> c9
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, nil) -> c12
   MIN(z0, cons(z1, z2)) -> c13(IF(le(z0, z1), min(z0, z2), min(z1, z2)), LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z1, z2))
   DEL(z0, nil) -> c16
   DEL(z0, cons(z1, z2)) -> c17(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), DEL(z0, z2))
K tuples:none
Defined Rule Symbols:   le_2, eq_2, if_3, minsort_1, min_2, del_2

Defined Pair Symbols:   LE_2, EQ_2, IF_3, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5, c6_1, c7, c8, c9, c10_1, c11_3, c12, c13_2, c14_2, c15_2, c16, c17_2, c18_2


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

(63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   LE(s(z0), 0) -> c1
   DEL(z0, nil) -> c16
   IF(true, z0, z1) -> c7
   MIN(z0, nil) -> c12
   IF(false, z0, z1) -> c8
   EQ(0, s(z0)) -> c4
   MINSORT(nil) -> c9
   EQ(s(z0), 0) -> c5
   LE(0, z0) -> c
   EQ(0, 0) -> c3

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   minsort(nil) -> nil
   minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1))))
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   del(z0, nil) -> nil
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(IF(le(z0, z1), min(z0, z2), min(z1, z2)), LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), DEL(z0, z2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(IF(le(z0, z1), min(z0, z2), min(z1, z2)), LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), DEL(z0, z2))
K tuples:none
Defined Rule Symbols:   le_2, eq_2, if_3, minsort_1, min_2, del_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c11_3, c13_2, c14_2, c15_2, c17_2, c18_2


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

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   minsort(nil) -> nil
   minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1))))
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   del(z0, nil) -> nil
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
K tuples:none
Defined Rule Symbols:   le_2, eq_2, if_3, minsort_1, min_2, del_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c11_3, c13_1, c14_1, c15_1, c17_1, c18_1


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

(67) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   minsort(nil) -> nil
   minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
K tuples:none
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c11_3, c13_1, c14_1, c15_1, c17_1, c18_1


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

(69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DEL(x_1, x_2)) = 0
   POL(EQ(x_1, x_2)) = 0
   POL(LE(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(MINSORT(x_1)) = [1]
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(del(x_1, x_2)) = [1] + x_1 + x_2
   POL(eq(x_1, x_2)) = 0
   POL(false) = 0
   POL(if(x_1, x_2, x_3)) = x_1 + x_3
   POL(le(x_1, x_2)) = x_1
   POL(min(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c11_3, c13_1, c14_1, c15_1, c17_1, c18_1


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) by 
   MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_2


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) by 
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)), MIN(x0, nil))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)), MIN(x0, nil))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)), MIN(x0, nil))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_2, c11_1


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_2, c11_1


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

(79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
We considered the (Usable) Rules:none
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DEL(x_1, x_2)) = 0
   POL(EQ(x_1, x_2)) = 0
   POL(LE(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(MINSORT(x_1)) = [1]
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(del(x_1, x_2)) = 0
   POL(eq(x_1, x_2)) = [1]
   POL(false) = [1]
   POL(if(x_1, x_2, x_3)) = 0
   POL(le(x_1, x_2)) = x_1 + x_2
   POL(min(x_1, x_2)) = 0
   POL(nil) = 0
   POL(s(x_1)) = [1] + x_1
   POL(true) = [1]

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_2, c11_1


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by 
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c11(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))), MIN(x0, cons(x1, x2)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c11(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))), MIN(x0, cons(x1, x2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c11(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))), MIN(x0, cons(x1, x2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_2, c11_3, c11_1


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

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_2, c11_3, c11_1, c_1


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

(85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
We considered the (Usable) Rules:none
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DEL(x_1, x_2)) = 0
   POL(EQ(x_1, x_2)) = 0
   POL(LE(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(MINSORT(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(cons(x_1, x_2)) = [3]
   POL(del(x_1, x_2)) = 0
   POL(eq(x_1, x_2)) = 0
   POL(false) = 0
   POL(if(x_1, x_2, x_3)) = 0
   POL(le(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = [1] + [3]x_2
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(true) = 0

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_2, c11_3, c11_1, c_1


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) by 
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_1, c11_2, c_1


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

(89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
We considered the (Usable) Rules:none
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DEL(x_1, x_2)) = 0
   POL(EQ(x_1, x_2)) = 0
   POL(LE(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(MINSORT(x_1)) = [2]
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(cons(x_1, x_2)) = [3]
   POL(del(x_1, x_2)) = [1] + [3]x_2
   POL(eq(x_1, x_2)) = 0
   POL(false) = 0
   POL(if(x_1, x_2, x_3)) = [1] + x_3
   POL(le(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(true) = 0

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_1, c11_2, c_1


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

(91) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_1, c11_2, c_1


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

(93) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_1, c11_2, c_1


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

(95) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_2, c11_3, c_1


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

(97) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_2, c11_3, c_1


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

(99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LE(s(z0), s(z1)) -> c2(LE(z0, z1)) by 
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
S tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, MIN_2, DEL_2, LE_2

Compound Symbols:   c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_2, c11_3, c_1, c2_1


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

(101) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
S tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, MIN_2, DEL_2, LE_2

Compound Symbols:   c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_2, c11_3, c_1, c2_1


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

(103) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
S tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, MIN_2, DEL_2, LE_2

Compound Symbols:   c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1


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

(105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) by 
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, MIN_2, DEL_2, LE_2, EQ_2

Compound Symbols:   c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1


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

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) by 
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1


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

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1))
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) by 
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1


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

(113) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) by MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1


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

(115) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1


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

(117) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) by MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1


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

(119) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) by MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) by MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) by MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c_1, c11_2, c11_3, c2_1, c6_1, c10_1, c13_1


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

(127) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c_1, c11_2, c11_3, c2_1, c6_1, c10_1, c13_1


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

(129) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c_1, c11_3, c11_2, c2_1, c6_1, c10_1, c13_1


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

(131) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) by MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c14_1, c15_1, c17_1, c18_1, c11_1, c_1, c11_3, c11_2, c2_1, c6_1, c10_1, c13_1


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

(133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) by 
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c18_1, c11_1, c_1, c11_3, c11_2, c2_1, c6_1, c10_1, c13_1, c14_1


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

(135) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c18_1, c11_1, c_1, c11_3, c11_2, c2_1, c6_1, c10_1, c13_1, c14_1


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

(137) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c18_1, c11_1, c_1, c11_3, c11_2, c2_1, c6_1, c10_1, c13_1, c14_1


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

(139) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c18_1, c11_1, c_1, c11_2, c2_1, c6_1, c10_1, c11_3, c13_1, c14_1


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

(141) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c18_1, c11_1, c_1, c2_1, c11_2, c6_1, c10_1, c11_3, c13_1, c14_1


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

(143) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
----------------------------------------

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c18_1, c11_1, c_1, c2_1, c11_2, c6_1, c10_1, c11_3, c13_1, c14_1


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

(145) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
S tuples:
   MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2))
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c18_1, c11_1, c_1, c2_1, c6_1, c10_1, c11_3, c13_1, c11_2, c14_1


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

(147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) by 
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
S tuples:
   DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1))
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   DEL_2, MINSORT_1, LE_2, EQ_2, MIN_2

Compound Symbols:   c17_1, c18_1, c11_1, c_1, c2_1, c6_1, c10_1, c11_3, c13_1, c11_2, c14_1, c15_1


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

(149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) by 
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
S tuples:
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   DEL_2, MINSORT_1, LE_2, EQ_2, MIN_2

Compound Symbols:   c18_1, c11_1, c_1, c2_1, c6_1, c10_1, c11_3, c13_1, c11_2, c14_1, c15_1, c17_1


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

(151) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
S tuples:
   DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   DEL_2, MINSORT_1, LE_2, EQ_2, MIN_2

Compound Symbols:   c18_1, c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1


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

(153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) by 
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2

Compound Symbols:   c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1


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

(155) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) by MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2

Compound Symbols:   c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1


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

(157) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) by MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2

Compound Symbols:   c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1


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

(159) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) by MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2

Compound Symbols:   c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) by MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2

Compound Symbols:   c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1


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

(163) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) by MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2

Compound Symbols:   c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1


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

(165) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) by MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2

Compound Symbols:   c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1


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

(167) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) by MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
----------------------------------------

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2

Compound Symbols:   c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_2, c14_1, c11_3, c15_1, c17_1, c18_1


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

(169) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2)))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2

Compound Symbols:   c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c14_1, c11_3, c11_2, c15_1, c17_1, c18_1


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

(171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) by 
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c(MIN(s(s(y0)), cons(s(s(y1)), z2)))
   MINSORT(cons(z0, cons(z1, cons(y2, y3)))) -> c(MIN(z0, cons(z1, cons(y2, y3))))
   MINSORT(cons(s(s(y0)), cons(z1, cons(s(s(y2)), y3)))) -> c(MIN(s(s(y0)), cons(z1, cons(s(s(y2)), y3))))
   MINSORT(cons(z0, cons(s(s(y1)), cons(s(s(y2)), y3)))) -> c(MIN(z0, cons(s(s(y1)), cons(s(s(y2)), y3))))
   MINSORT(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c(MIN(z0, cons(z1, cons(y2, cons(y3, y4)))))
   MINSORT(cons(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4))))) -> c(MIN(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4)))))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c(MIN(s(s(y0)), cons(s(s(y1)), z2)))
   MINSORT(cons(z0, cons(z1, cons(y2, y3)))) -> c(MIN(z0, cons(z1, cons(y2, y3))))
   MINSORT(cons(s(s(y0)), cons(z1, cons(s(s(y2)), y3)))) -> c(MIN(s(s(y0)), cons(z1, cons(s(s(y2)), y3))))
   MINSORT(cons(z0, cons(s(s(y1)), cons(s(s(y2)), y3)))) -> c(MIN(z0, cons(s(s(y1)), cons(s(s(y2)), y3))))
   MINSORT(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c(MIN(z0, cons(z1, cons(y2, cons(y3, y4)))))
   MINSORT(cons(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4))))) -> c(MIN(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4)))))
S tuples:
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1))))
   MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2)))
   MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil)))
   MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2)))
   MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3))))
   MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2)))
   DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil)))
   MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3))))
   MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
K tuples:
   MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil)))
   MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c(MIN(s(s(y0)), cons(s(s(y1)), z2)))
   MINSORT(cons(z0, cons(z1, cons(y2, y3)))) -> c(MIN(z0, cons(z1, cons(y2, y3))))
   MINSORT(cons(s(s(y0)), cons(z1, cons(s(s(y2)), y3)))) -> c(MIN(s(s(y0)), cons(z1, cons(s(s(y2)), y3))))
   MINSORT(cons(z0, cons(s(s(y1)), cons(s(s(y2)), y3)))) -> c(MIN(z0, cons(s(s(y1)), cons(s(s(y2)), y3))))
   MINSORT(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c(MIN(z0, cons(z1, cons(y2, cons(y3, y4)))))
   MINSORT(cons(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4))))) -> c(MIN(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4)))))
Defined Rule Symbols:   del_2, min_2, if_3, le_2, eq_2

Defined Pair Symbols:   MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2

Compound Symbols:   c11_1, c2_1, c6_1, c10_1, c13_1, c_1, c14_1, c11_3, c11_2, c15_1, c17_1, c18_1