KILLED
proof of input_j7MptV71Tn.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (12) CpxRNTS
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
        (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 333 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 132 ms]
        (28) CpxRNTS
        (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 388 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
        (34) CpxRNTS
        (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 1173 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 659 ms]
        (40) CpxRNTS
        (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 1050 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 457 ms]
        (46) CpxRNTS
        (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 1158 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 277 ms]
        (52) CpxRNTS
(53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(54) CdtProblem
    (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 169 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtLeafRemovalProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 79 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 69 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRhsSimplificationProcessorProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtForwardInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) 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)
   if1(true, x, y, xs) -> min(x, xs)
   if1(false, x, y, xs) -> min(y, xs)
   if2(true, x, y, xs) -> xs
   if2(false, x, y, xs) -> cons(y, del(x, xs))
   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)) -> if1(le(x, y), x, y, z)
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if2(eq(x, y), x, y, 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)
   if1(true, x, y, xs) -> min(x, xs)
   if1(false, x, y, xs) -> min(y, xs)
   if2(true, x, y, xs) -> xs
   if2(false, x, y, xs) -> cons(y, del(x, xs))
   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)) -> if1(le(x, y), x, y, z)
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if2(eq(x, y), x, y, 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)
   if1(true, x, y, xs) -> min(x, xs)
   if1(false, x, y, xs) -> min(y, xs)
   if2(true, x, y, xs) -> xs
   if2(false, x, y, xs) -> cons(y, del(x, xs))
   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)) -> if1(le(x, y), x, y, z)
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if2(eq(x, y), x, y, 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]
   if1(true, x, y, xs) -> min(x, xs) [1]
   if1(false, x, y, xs) -> min(y, xs) [1]
   if2(true, x, y, xs) -> xs [1]
   if2(false, x, y, xs) -> cons(y, del(x, xs)) [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)) -> if1(le(x, y), x, y, z) [1]
   del(x, nil) -> nil [1]
   del(x, cons(y, z)) -> if2(eq(x, y), x, y, 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]
   if1(true, x, y, xs) -> min(x, xs) [1]
   if1(false, x, y, xs) -> min(y, xs) [1]
   if2(true, x, y, xs) -> xs [1]
   if2(false, x, y, xs) -> cons(y, del(x, xs)) [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)) -> if1(le(x, y), x, y, z) [1]
   del(x, nil) -> nil [1]
   del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) [1]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
eq :: 0:s -> 0:s -> true:false
if1 :: true:false -> 0:s -> 0:s -> cons:nil -> 0:s
min :: 0:s -> cons:nil -> 0:s
if2 :: true:false -> 0:s -> 0:s -> cons:nil -> cons:nil
cons :: 0:s -> cons:nil -> cons:nil
del :: 0:s -> cons:nil -> cons:nil
minsort :: cons:nil -> cons:nil
nil :: cons:nil

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

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

And the following fresh constants: none

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

(10)
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]
   if1(true, x, y, xs) -> min(x, xs) [1]
   if1(false, x, y, xs) -> min(y, xs) [1]
   if2(true, x, y, xs) -> xs [1]
   if2(false, x, y, xs) -> cons(y, del(x, xs)) [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)) -> if1(le(x, y), x, y, z) [1]
   del(x, nil) -> nil [1]
   del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) [1]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
eq :: 0:s -> 0:s -> true:false
if1 :: true:false -> 0:s -> 0:s -> cons:nil -> 0:s
min :: 0:s -> cons:nil -> 0:s
if2 :: true:false -> 0:s -> 0:s -> cons:nil -> cons:nil
cons :: 0:s -> cons:nil -> cons:nil
del :: 0:s -> cons:nil -> cons:nil
minsort :: cons:nil -> cons:nil
nil :: cons:nil

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

(11) 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 => 1
   false => 0
   nil => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   del(z', z'') -{ 1 }-> if2(eq(x, y), x, y, z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
   del(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
   eq(z', z'') -{ 1 }-> eq(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: y >= 0, z'' = 1 + y, z' = 0
   eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 1 + x, x >= 0
   if1(z', z'', z1, z2) -{ 1 }-> min(x, xs) :|: xs >= 0, z1 = y, x >= 0, y >= 0, z'' = x, z' = 1, z2 = xs
   if1(z', z'', z1, z2) -{ 1 }-> min(y, xs) :|: xs >= 0, z1 = y, x >= 0, y >= 0, z'' = x, z2 = xs, z' = 0
   if2(z', z'', z1, z2) -{ 1 }-> xs :|: xs >= 0, z1 = y, x >= 0, y >= 0, z'' = x, z' = 1, z2 = xs
   if2(z', z'', z1, z2) -{ 1 }-> 1 + y + del(x, xs) :|: xs >= 0, z1 = y, x >= 0, y >= 0, z'' = x, z2 = xs, z' = 0
   le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   le(z', z'') -{ 1 }-> 1 :|: z'' = y, y >= 0, z' = 0
   le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 1 + x, x >= 0
   min(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   min(z', z'') -{ 1 }-> if1(le(x, y), x, y, z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
   minsort(z') -{ 1 }-> 0 :|: z' = 0
   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.

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

(13) 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:

   del_2
   min_2
   eq_2
   le_2
   if2_4
   if1_4

Due to the following rules being added:
none

And the following fresh constants: none

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

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

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

   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]
   if1(true, x, y, xs) -> min(x, xs) [1]
   if1(false, x, y, xs) -> min(y, xs) [1]
   if2(true, x, y, xs) -> xs [1]
   if2(false, x, y, xs) -> cons(y, del(x, xs)) [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)) -> if1(le(x, y), x, y, z) [1]
   del(x, nil) -> nil [1]
   del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) [1]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
eq :: 0:s -> 0:s -> true:false
if1 :: true:false -> 0:s -> 0:s -> cons:nil -> 0:s
min :: 0:s -> cons:nil -> 0:s
if2 :: true:false -> 0:s -> 0:s -> cons:nil -> cons:nil
cons :: 0:s -> cons:nil -> cons:nil
del :: 0:s -> cons:nil -> cons:nil
minsort :: cons:nil -> cons:nil
nil :: cons:nil

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

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

(16)
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]
   if1(true, x, y, xs) -> min(x, xs) [1]
   if1(false, x, y, xs) -> min(y, xs) [1]
   if2(true, x, y, xs) -> xs [1]
   if2(false, x, y, xs) -> cons(y, del(x, xs)) [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(if1(le(x, y'), x, y', z'), cons(x, cons(y', z'))))) [2]
   min(x, nil) -> x [1]
   min(0, cons(y, z)) -> if1(true, 0, y, z) [2]
   min(s(x'), cons(0, z)) -> if1(false, s(x'), 0, z) [2]
   min(s(x''), cons(s(y''), z)) -> if1(le(x'', y''), s(x''), s(y''), z) [2]
   del(x, nil) -> nil [1]
   del(0, cons(0, z)) -> if2(true, 0, 0, z) [2]
   del(0, cons(s(y1), z)) -> if2(false, 0, s(y1), z) [2]
   del(s(x1), cons(0, z)) -> if2(false, s(x1), 0, z) [2]
   del(s(x2), cons(s(y2), z)) -> if2(eq(x2, y2), s(x2), s(y2), z) [2]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
eq :: 0:s -> 0:s -> true:false
if1 :: true:false -> 0:s -> 0:s -> cons:nil -> 0:s
min :: 0:s -> cons:nil -> 0:s
if2 :: true:false -> 0:s -> 0:s -> cons:nil -> cons:nil
cons :: 0:s -> cons:nil -> cons:nil
del :: 0:s -> cons:nil -> cons:nil
minsort :: cons:nil -> cons:nil
nil :: cons:nil

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

(17) 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 => 1
   false => 0
   nil => 0

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

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

   del(z'', z1) -{ 2 }-> if2(eq(x2, y2), 1 + x2, 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, z'' = 1 + x2, y2 >= 0, x2 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z) :|: z'' = 0, z >= 0, z1 = 1 + 0 + z
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + x1, 0, z) :|: x1 >= 0, z >= 0, z'' = 1 + x1, z1 = 1 + 0 + z
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 0, z'' = x
   eq(z'', z1) -{ 1 }-> eq(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, y >= 0, z1 = 1 + y
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 0, z'' = 1 + x
   if1(z'', z1, z2, z3) -{ 1 }-> min(x, xs) :|: xs >= 0, z2 = y, x >= 0, y >= 0, z3 = xs, z'' = 1, z1 = x
   if1(z'', z1, z2, z3) -{ 1 }-> min(y, xs) :|: z'' = 0, xs >= 0, z2 = y, x >= 0, y >= 0, z3 = xs, z1 = x
   if2(z'', z1, z2, z3) -{ 1 }-> xs :|: xs >= 0, z2 = y, x >= 0, y >= 0, z3 = xs, z'' = 1, z1 = x
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + y + del(x, xs) :|: z'' = 0, xs >= 0, z2 = y, x >= 0, y >= 0, z3 = xs, z1 = x
   le(z'', z1) -{ 1 }-> le(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = y, y >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 0, z'' = 1 + x
   min(z'', z1) -{ 1 }-> x :|: z1 = 0, x >= 0, z'' = x
   min(z'', z1) -{ 2 }-> if1(le(x'', y''), 1 + x'', 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' = 1 + x'', x'' >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + x', 0, z) :|: z >= 0, z'' = 1 + x', x' >= 0, z1 = 1 + 0 + z
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 2 }-> 1 + min(x, 0) + minsort(del(x, 1 + x + 0)) :|: z'' = 1 + x + 0, x >= 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, y', z'), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0


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

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

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

   del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 2 }-> if1(le(z'' - 1, y''), 1 + (z'' - 1), 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, 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'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0


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

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

   { le }
   { eq }
   { min, if1 }
   { del, if2 }
   { minsort }

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

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

   del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 2 }-> if1(le(z'' - 1, y''), 1 + (z'' - 1), 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, 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'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0

Function symbols to be analyzed: {le}, {eq}, {min,if1}, {del,if2}, {minsort}

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

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

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

   del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 2 }-> if1(le(z'' - 1, y''), 1 + (z'' - 1), 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, 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'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0

Function symbols to be analyzed: {le}, {eq}, {min,if1}, {del,if2}, {minsort}

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

(25) 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: 1

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

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

   del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 2 }-> if1(le(z'' - 1, y''), 1 + (z'' - 1), 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, 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'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0

Function symbols to be analyzed: {le}, {eq}, {min,if1}, {del,if2}, {minsort}
Previous analysis results are:
  le: runtime: ?, size: O(1) [1]

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

(27) 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

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

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

   del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 2 }-> if1(le(z'' - 1, y''), 1 + (z'' - 1), 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, 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'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0

Function symbols to be analyzed: {eq}, {min,if1}, {del,if2}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [1]

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

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

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

   del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0

Function symbols to be analyzed: {eq}, {min,if1}, {del,if2}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [1]

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

(31) 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: 1

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

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

   del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0

Function symbols to be analyzed: {eq}, {min,if1}, {del,if2}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [1]
  eq: runtime: ?, size: O(1) [1]

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

(33) 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 + z1

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

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

   del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0

Function symbols to be analyzed: {min,if1}, {del,if2}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z1], size: O(1) [1]

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

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

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

   del(z'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0

Function symbols to be analyzed: {min,if1}, {del,if2}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z1], size: O(1) [1]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: if1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z1 + z2 + z3

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

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

   del(z'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0

Function symbols to be analyzed: {min,if1}, {del,if2}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z1], size: O(1) [1]
  min: runtime: ?, size: O(n^1) [z'' + z1]
  if1: runtime: ?, size: O(n^1) [z1 + z2 + z3]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: min
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 20 + z'' + 7*z1

Computed RUNTIME bound using CoFloCo for: if1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 21 + z1 + z2 + 7*z3

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

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

   del(z'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0
   minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0

Function symbols to be analyzed: {del,if2}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z1], size: O(1) [1]
  min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1]
  if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3]

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

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

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

   del(z'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 21 + z'' }-> 1 + s5 + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0
   minsort(z'') -{ 52 + 2*x + 9*y' + 14*z' }-> 1 + s6 + minsort(del(s7, 1 + x + (1 + y' + z'))) :|: s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0

Function symbols to be analyzed: {del,if2}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z1], size: O(1) [1]
  min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1]
  if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z2 + z3

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

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

   del(z'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 21 + z'' }-> 1 + s5 + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0
   minsort(z'') -{ 52 + 2*x + 9*y' + 14*z' }-> 1 + s6 + minsort(del(s7, 1 + x + (1 + y' + z'))) :|: s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0

Function symbols to be analyzed: {del,if2}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z1], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z1], size: O(1) [1]
  min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1]
  if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3]
  del: runtime: ?, size: O(n^1) [z1]
  if2: runtime: ?, size: O(n^1) [1 + z2 + z3]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: del
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 49 + 17*z1

Computed RUNTIME bound using CoFloCo for: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 50 + 17*z3

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

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

   del(z'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 21 + z'' }-> 1 + s5 + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0
   minsort(z'') -{ 52 + 2*x + 9*y' + 14*z' }-> 1 + s6 + minsort(del(s7, 1 + x + (1 + y' + z'))) :|: s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, 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) [1]
  eq: runtime: O(n^1) [3 + z1], size: O(1) [1]
  min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1]
  if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3]
  del: runtime: O(n^1) [49 + 17*z1], size: O(n^1) [z1]
  if2: runtime: O(n^1) [50 + 17*z3], size: O(n^1) [1 + z2 + z3]

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

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

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

   del(z'', z1) -{ 35 + 17*z1 }-> s14 :|: s14 >= 0, s14 <= 0 + (z1 - 1) + 1, z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 52 + 17*z }-> s15 :|: s15 >= 0, s15 <= 1 + y1 + z + 1, z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 35 + 17*z1 }-> s16 :|: s16 >= 0, s16 <= 0 + (z1 - 1) + 1, z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 55 + y2 + 17*z }-> s17 :|: s17 >= 0, s17 <= 1 + y2 + z + 1, s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 50 + 17*z3 }-> 1 + z2 + s11 :|: s11 >= 0, s11 <= z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 70 + 18*z'' }-> 1 + s5 + minsort(s12) :|: s12 >= 0, s12 <= 1 + (z'' - 1) + 0, s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0
   minsort(z'') -{ 135 + 19*x + 26*y' + 31*z' }-> 1 + s6 + minsort(s13) :|: s13 >= 0, s13 <= 1 + x + (1 + y' + z'), s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, 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) [1]
  eq: runtime: O(n^1) [3 + z1], size: O(1) [1]
  min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1]
  if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3]
  del: runtime: O(n^1) [49 + 17*z1], size: O(n^1) [z1]
  if2: runtime: O(n^1) [50 + 17*z3], size: O(n^1) [1 + z2 + z3]

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

(49) 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: ?

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

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

   del(z'', z1) -{ 35 + 17*z1 }-> s14 :|: s14 >= 0, s14 <= 0 + (z1 - 1) + 1, z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 52 + 17*z }-> s15 :|: s15 >= 0, s15 <= 1 + y1 + z + 1, z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 35 + 17*z1 }-> s16 :|: s16 >= 0, s16 <= 0 + (z1 - 1) + 1, z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 55 + y2 + 17*z }-> s17 :|: s17 >= 0, s17 <= 1 + y2 + z + 1, s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 50 + 17*z3 }-> 1 + z2 + s11 :|: s11 >= 0, s11 <= z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 70 + 18*z'' }-> 1 + s5 + minsort(s12) :|: s12 >= 0, s12 <= 1 + (z'' - 1) + 0, s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0
   minsort(z'') -{ 135 + 19*x + 26*y' + 31*z' }-> 1 + s6 + minsort(s13) :|: s13 >= 0, s13 <= 1 + x + (1 + y' + z'), s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, 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) [1]
  eq: runtime: O(n^1) [3 + z1], size: O(1) [1]
  min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1]
  if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3]
  del: runtime: O(n^1) [49 + 17*z1], size: O(n^1) [z1]
  if2: runtime: O(n^1) [50 + 17*z3], size: O(n^1) [1 + z2 + z3]
  minsort: runtime: ?, size: INF

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

(51) 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: ?

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

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

   del(z'', z1) -{ 35 + 17*z1 }-> s14 :|: s14 >= 0, s14 <= 0 + (z1 - 1) + 1, z'' = 0, z1 - 1 >= 0
   del(z'', z1) -{ 52 + 17*z }-> s15 :|: s15 >= 0, s15 <= 1 + y1 + z + 1, z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z
   del(z'', z1) -{ 35 + 17*z1 }-> s16 :|: s16 >= 0, s16 <= 0 + (z1 - 1) + 1, z'' - 1 >= 0, z1 - 1 >= 0
   del(z'', z1) -{ 55 + y2 + 17*z }-> s17 :|: s17 >= 0, s17 <= 1 + y2 + z + 1, s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0
   del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   if2(z'', z1, z2, z3) -{ 50 + 17*z3 }-> 1 + z2 + s11 :|: s11 >= 0, s11 <= z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
   le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0
   le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0
   le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0
   min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0
   min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0
   minsort(z'') -{ 1 }-> 0 :|: z'' = 0
   minsort(z'') -{ 70 + 18*z'' }-> 1 + s5 + minsort(s12) :|: s12 >= 0, s12 <= 1 + (z'' - 1) + 0, s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0
   minsort(z'') -{ 135 + 19*x + 26*y' + 31*z' }-> 1 + s6 + minsort(s13) :|: s13 >= 0, s13 <= 1 + x + (1 + y' + z'), s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, 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) [1]
  eq: runtime: O(n^1) [3 + z1], size: O(1) [1]
  min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1]
  if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3]
  del: runtime: O(n^1) [49 + 17*z1], size: O(n^1) [z1]
  if2: runtime: O(n^1) [50 + 17*z3], size: O(n^1) [1 + z2 + z3]
  minsort: runtime: INF, size: INF

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

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

(54)
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)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> if1(le(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, 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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(true, z0, z1, z2) -> c9
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(nil) -> c11
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, nil) -> c14
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, nil) -> c16
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(true, z0, z1, z2) -> c9
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(nil) -> c11
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, nil) -> c14
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, nil) -> c16
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
K tuples:none
Defined Rule Symbols:   le_2, eq_2, if1_4, if2_4, minsort_1, min_2, del_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5, c6_1, c7_1, c8_1, c9, c10_1, c11, c12_1, c13_3, c14, c15_2, c16, c17_2


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

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 9 trailing nodes:
   MINSORT(nil) -> c11
   EQ(s(z0), 0) -> c5
   EQ(0, 0) -> c3
   LE(s(z0), 0) -> c1
   EQ(0, s(z0)) -> c4
   DEL(z0, nil) -> c16
   IF2(true, z0, z1, z2) -> c9
   MIN(z0, nil) -> c14
   LE(0, z0) -> c

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

(56)
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)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> if1(le(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
K tuples:none
Defined Rule Symbols:   le_2, eq_2, if1_4, if2_4, minsort_1, min_2, del_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c15_2, c17_2


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

(57) 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))))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
K tuples:none
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c15_2, c17_2


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

(59) 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)) -> c12(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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
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(IF1(x_1, x_2, x_3, x_4)) = 0
   POL(IF2(x_1, x_2, x_3, x_4)) = 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(c12(x_1)) = x_1
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(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)) = [1]
   POL(false) = [1]
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(le(x_1, x_2)) = [1]
   POL(min(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = [1]

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c15_2, c17_2


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) by 
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_2, c17_2, c13_3


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

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_2, c17_2, c13_3, c13_2


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) by 
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2), LE(0, z0))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2), LE(s(z0), 0))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2), LE(0, z0))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2), LE(s(z0), 0))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2), LE(0, z0))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2), LE(s(z0), 0))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, DEL_2, MIN_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c17_2, c13_3, c13_2, c15_2


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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, DEL_2, MIN_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c17_2, c13_3, c13_2, c15_2, c15_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) by 
   DEL(0, cons(0, x2)) -> c17(IF2(true, 0, 0, x2), EQ(0, 0))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2), EQ(0, s(z0)))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2), EQ(s(z0), 0))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(0, x2)) -> c17(IF2(true, 0, 0, x2), EQ(0, 0))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2), EQ(0, s(z0)))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2), EQ(s(z0), 0))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(0, x2)) -> c17(IF2(true, 0, 0, x2), EQ(0, 0))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2), EQ(0, s(z0)))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2), EQ(s(z0), 0))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c13_2, c15_2, c15_1, c17_2


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

(71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   DEL(0, cons(0, x2)) -> c17(IF2(true, 0, 0, x2), EQ(0, 0))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2), EQ(0, s(z0)))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2), EQ(s(z0), 0))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2), EQ(0, s(z0)))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2), EQ(s(z0), 0))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c13_2, c15_2, c15_1, c17_2


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c13_2, c15_2, c15_1, c17_2, c17_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) by 
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), 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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), 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)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c13_2, c15_2, c15_1, c17_2, c17_1


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), 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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c13_2, c15_2, c15_1, c17_2, c17_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_2, c15_2, c15_1, c17_2, c17_1, c13_3, c13_1


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

(81) 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(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(IF1(x_1, x_2, x_3, x_4)) = 0
   POL(IF2(x_1, x_2, x_3, x_4)) = x_3
   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(c12(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c15(x_1)) = x_1
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c17(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(del(x_1, x_2)) = [1] + x_2
   POL(eq(x_1, x_2)) = 0
   POL(false) = 0
   POL(if1(x_1, x_2, x_3, x_4)) = 0
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4
   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

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_2, c15_2, c15_1, c17_2, c17_1, c13_3, c13_1


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) by 
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_2, c15_1, c17_2, c17_1, c13_3, c13_2, c13_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, nil)) -> c13(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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(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(IF1(x_1, x_2, x_3, x_4)) = 0
   POL(IF2(x_1, x_2, x_3, x_4)) = x_3
   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(c12(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c15(x_1)) = x_1
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c17(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(del(x_1, x_2)) = x_2
   POL(eq(x_1, x_2)) = 0
   POL(false) = 0
   POL(if1(x_1, x_2, x_3, x_4)) = 0
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4
   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

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_2, c15_1, c17_2, c17_1, c13_3, c13_2, c13_1


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) by 
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_1, c17_2, c17_1, c13_3, c13_2, c13_1, c15_2


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) by 
   DEL(s(0), cons(s(0), x2)) -> c17(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_2, c13_1, c15_2, c17_2


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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(true, z0, z1, z2) -> c7(MIN(z0, z2))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_2, c13_1, c15_2, c17_2


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

(93) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) by 
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_2, c13_1, c15_2, c17_2, c7_1


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

(95) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, 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))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, 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))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_2, c13_1, c15_2, c17_2, c7_1


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

(97) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, 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))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_2, c13_3, c13_1, c15_2, c17_2, c7_1


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

(99) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, 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))
   IF1(false, z0, z1, z2) -> c8(MIN(z1, z2))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2

Compound Symbols:   c2_1, c6_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_2, c13_3, c13_1, c15_2, c17_2, c7_1


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

(101) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) by 
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4

Compound Symbols:   c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_2, c13_3, c13_1, c15_2, c17_2, c7_1, c8_1


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

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4

Compound Symbols:   c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_2, c13_3, c13_1, c15_2, c17_2, c7_1, c8_1, c_1


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

(105) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), 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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2)))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4

Compound Symbols:   c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_1, c13_2, c15_2, c17_2, c7_1, c8_1, c_1


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

(107) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, 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))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, 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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4

Compound Symbols:   c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_1, c13_2, c15_2, c17_2, c7_1, c8_1, c_1


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

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, 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))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4

Compound Symbols:   c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_1, c13_2, c15_2, c17_2, c7_1, c8_1, c_1


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

(111) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 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))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4

Compound Symbols:   c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_1, c13_2, c15_2, c17_2, c7_1, c8_1, c_1


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

(113) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4

Compound Symbols:   c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_1, c13_2, c15_2, c17_2, c7_1, c13_3, c8_1, c_1


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

(115) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4

Compound Symbols:   c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_2, c13_1, c15_2, c17_2, c7_1, c13_3, c8_1, c_1


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

(117) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, 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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4

Compound Symbols:   c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1


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

(119) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) by MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(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))
   IF2(false, z0, z1, z2) -> c10(DEL(z0, z2))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4

Compound Symbols:   c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1


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

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

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4

Compound Symbols:   c2_1, c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1, c10_1


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

(123) 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)))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   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(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
   MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2

Compound Symbols:   c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1, c10_1, c2_1


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

(125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0)))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   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(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2

Compound Symbols:   c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1, c10_1, c2_1


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

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
S tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2

Compound Symbols:   c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1, c10_1, c2_1


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

(129) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
S tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2

Compound Symbols:   c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1, c10_1, c2_1


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

(131) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
S tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2

Compound Symbols:   c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_2, c8_1, c_1, c13_3, c13_1, c10_1, c2_1


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

(133) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
S tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2

Compound Symbols:   c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_2, c13_3, c13_1, c10_1, c2_1


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

(135) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
S tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2

Compound Symbols:   c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1


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

(137) 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)))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
S tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2, EQ_2

Compound Symbols:   c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1


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

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   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)) -> c12(MIN(z0, z1))
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
S tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
K tuples:
   MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2, EQ_2

Compound Symbols:   c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17


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

(141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) by 
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
S tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1


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

(143) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) by MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
----------------------------------------

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
S tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1


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

(145) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) by MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
S tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1)))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1


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

(147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) by 
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
S tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1


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

(149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1))) 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)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
S tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, 0, x0, x1) -> c7(MIN(0, x1))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1


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

(151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) by 
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
S tuples:
   MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2))
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1


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

(153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) by 
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
S tuples:
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1


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

(155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) by 
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
S tuples:
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1


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

(157) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), 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))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
S tuples:
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1


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

(159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) by 
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
S tuples:
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1, c7_1


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

(161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) by 
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
S tuples:
   MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2))
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c15_1, c17_1, c15_2, c17_2, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1, c7_1


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

(163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) by 
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
S tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   DEL_2, MIN_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c17_1, c15_2, c17_2, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1


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

(165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) by 
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
S tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   DEL_2, MIN_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2

Compound Symbols:   c17_1, c15_2, c17_2, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1


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

(167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) by 
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))
S tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   DEL_2, MIN_2, MINSORT_1, IF2_4, LE_2, EQ_2, IF1_4

Compound Symbols:   c17_1, c15_2, c17_2, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1, c8_1


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

(169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) by 
   MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2)))
   MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))
   MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2)))
   MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))))
S tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))
   MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2)))
   MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   DEL_2, MIN_2, MINSORT_1, IF2_4, LE_2, EQ_2, IF1_4

Compound Symbols:   c17_1, c15_2, c17_2, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1, c8_1, c_1


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

(171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) by 
   MINSORT(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil)))
   MINSORT(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil)))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))
   MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2)))
   MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))))
   MINSORT(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil)))
   MINSORT(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil)))
S tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))
   MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2)))
   MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   DEL_2, MIN_2, MINSORT_1, IF2_4, LE_2, EQ_2, IF1_4

Compound Symbols:   c17_1, c15_2, c17_2, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1, c8_1, c_1


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

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

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))
   MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2)))
   MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))))
   MINSORT(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil)))
   MINSORT(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil)))
   IF2(false, 0, s(z0), cons(s(y0), y1)) -> c10(DEL(0, cons(s(y0), y1)))
S tuples:
   DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2))
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))
   MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2)))
   MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF2(false, 0, s(z0), cons(s(y0), y1)) -> c10(DEL(0, cons(s(y0), y1)))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   DEL_2, MIN_2, MINSORT_1, IF2_4, LE_2, EQ_2, IF1_4

Compound Symbols:   c17_1, c15_2, c17_2, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1, c8_1, c_1


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

(175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) by 
   DEL(0, cons(s(z0), cons(s(y1), y2))) -> c17(IF2(false, 0, s(z0), cons(s(y1), y2)))

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(z0, z2)
   if1(false, z0, z1, z2) -> min(z1, z2)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> z2
   if2(false, z0, z1, z2) -> cons(z1, del(z0, z2))
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
Tuples:
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))
   MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2)))
   MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))))
   MINSORT(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil)))
   MINSORT(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil)))
   IF2(false, 0, s(z0), cons(s(y0), y1)) -> c10(DEL(0, cons(s(y0), y1)))
   DEL(0, cons(s(z0), cons(s(y1), y2))) -> c17(IF2(false, 0, s(z0), cons(s(y1), y2)))
S tuples:
   DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2))
   MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1))
   IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2))
   DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2))
   DEL(s(0), cons(s(0), x2)) -> c17
   MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1)))
   MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1)))
   MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1))))
   DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1)))
   MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1)))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1)))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2)))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1)))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2))))
   MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1)))
   IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2)))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1)))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))
   MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2)))
   MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))))
   IF2(false, 0, s(z0), cons(s(y0), y1)) -> c10(DEL(0, cons(s(y0), y1)))
   DEL(0, cons(s(z0), cons(s(y1), y2))) -> c17(IF2(false, 0, s(z0), cons(s(y1), y2)))
K tuples:
   MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1)))
   MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1)))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2)))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2)))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1)))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1)))
Defined Rule Symbols:   del_2, min_2, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   DEL_2, MIN_2, MINSORT_1, IF2_4, LE_2, EQ_2, IF1_4

Compound Symbols:   c17_1, c15_2, c17_2, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1, c8_1, c_1