KILLED
proof of input_DTXjXGLI28.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 6 ms]
        (20) CpxRNTS
        (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) ResultPropagationProof [UPPER BOUND(ID), 2 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 343 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 161 ms]
        (28) CpxRNTS
        (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 144 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 168 ms]
        (34) CpxRNTS
        (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 718 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 415 ms]
        (40) CpxRNTS
        (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 2176 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 1054 ms]
        (46) CpxRNTS
        (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 1365 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 165 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)), 181 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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 78 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 100 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRewritingProof [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), 4 ms]
    (168) CdtProblem
    (169) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) 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)
   minsort(nil) -> nil
   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
   min(nil) -> 0
   min(cons(x, nil)) -> x
   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
   if1(true, x, y, xs) -> min(cons(x, xs))
   if1(false, x, y, xs) -> min(cons(y, xs))
   rm(x, nil) -> nil
   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
   if2(true, x, y, xs) -> rm(x, xs)
   if2(false, x, y, xs) -> cons(y, rm(x, xs))

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)
   minsort(nil) -> nil
   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
   min(nil) -> 0'
   min(cons(x, nil)) -> x
   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
   if1(true, x, y, xs) -> min(cons(x, xs))
   if1(false, x, y, xs) -> min(cons(y, xs))
   rm(x, nil) -> nil
   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
   if2(true, x, y, xs) -> rm(x, xs)
   if2(false, x, y, xs) -> cons(y, rm(x, xs))

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)
   minsort(nil) -> nil
   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
   min(nil) -> 0
   min(cons(x, nil)) -> x
   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
   if1(true, x, y, xs) -> min(cons(x, xs))
   if1(false, x, y, xs) -> min(cons(y, xs))
   rm(x, nil) -> nil
   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
   if2(true, x, y, xs) -> rm(x, xs)
   if2(false, x, y, xs) -> cons(y, rm(x, xs))

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]
   minsort(nil) -> nil [1]
   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) [1]
   min(nil) -> 0 [1]
   min(cons(x, nil)) -> x [1]
   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) [1]
   if1(true, x, y, xs) -> min(cons(x, xs)) [1]
   if1(false, x, y, xs) -> min(cons(y, xs)) [1]
   rm(x, nil) -> nil [1]
   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1]
   if2(true, x, y, xs) -> rm(x, xs) [1]
   if2(false, x, y, xs) -> cons(y, rm(x, xs)) [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]
   minsort(nil) -> nil [1]
   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) [1]
   min(nil) -> 0 [1]
   min(cons(x, nil)) -> x [1]
   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) [1]
   if1(true, x, y, xs) -> min(cons(x, xs)) [1]
   if1(false, x, y, xs) -> min(cons(y, xs)) [1]
   rm(x, nil) -> nil [1]
   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1]
   if2(true, x, y, xs) -> rm(x, xs) [1]
   if2(false, x, y, xs) -> cons(y, rm(x, xs)) [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
minsort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
min :: nil:cons -> 0:s
rm :: 0:s -> nil:cons -> nil:cons
if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s
if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons

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]
   minsort(nil) -> nil [1]
   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) [1]
   min(nil) -> 0 [1]
   min(cons(x, nil)) -> x [1]
   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) [1]
   if1(true, x, y, xs) -> min(cons(x, xs)) [1]
   if1(false, x, y, xs) -> min(cons(y, xs)) [1]
   rm(x, nil) -> nil [1]
   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1]
   if2(true, x, y, xs) -> rm(x, xs) [1]
   if2(false, x, y, xs) -> cons(y, rm(x, xs)) [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
minsort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
min :: nil:cons -> 0:s
rm :: 0:s -> nil:cons -> nil:cons
if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s
if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons

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:

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

   rm_2
   min_1
   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]
   minsort(nil) -> nil [1]
   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) [1]
   min(nil) -> 0 [1]
   min(cons(x, nil)) -> x [1]
   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) [1]
   if1(true, x, y, xs) -> min(cons(x, xs)) [1]
   if1(false, x, y, xs) -> min(cons(y, xs)) [1]
   rm(x, nil) -> nil [1]
   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1]
   if2(true, x, y, xs) -> rm(x, xs) [1]
   if2(false, x, y, xs) -> cons(y, rm(x, xs)) [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
minsort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
min :: nil:cons -> 0:s
rm :: 0:s -> nil:cons -> nil:cons
if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s
if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons

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

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:

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

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


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

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

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

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

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

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

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

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

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

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

Function symbols to be analyzed: {le}, {eq}, {min,if1}, {if2,rm}, {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 + z'

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: min
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 12 + 5*z + z^2

Computed RUNTIME bound using KoAT for: if1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2

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

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

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

Function symbols to be analyzed: {if2,rm}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z]
  if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1]

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

(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:

   eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0
   min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs)
   min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs)
   min(z) -{ 1 }-> 0 :|: z = 0
   min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   minsort(z) -{ 14 + 5*z + z^2 }-> 1 + s3 + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0
   minsort(z) -{ 68 + 16*x + 4*x*xs' + 2*x*y' + 2*x^2 + 23*xs' + 4*xs'*y' + 3*xs'^2 + 17*y' + 2*y'^2 }-> 1 + s4 + minsort(rm(s5, 1 + x + (1 + y' + xs'))) :|: s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   rm(z, z') -{ 5 + y2 }-> if2(s2, 1 + (z - 1), 1 + y2, xs) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0
   rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0
   rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {if2,rm}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z]
  if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0
   min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs)
   min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs)
   min(z) -{ 1 }-> 0 :|: z = 0
   min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   minsort(z) -{ 14 + 5*z + z^2 }-> 1 + s3 + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0
   minsort(z) -{ 68 + 16*x + 4*x*xs' + 2*x*y' + 2*x^2 + 23*xs' + 4*xs'*y' + 3*xs'^2 + 17*y' + 2*y'^2 }-> 1 + s4 + minsort(rm(s5, 1 + x + (1 + y' + xs'))) :|: s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   rm(z, z') -{ 5 + y2 }-> if2(s2, 1 + (z - 1), 1 + y2, xs) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0
   rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0
   rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {if2,rm}, {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z]
  if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1]
  if2: runtime: ?, size: O(n^1) [1 + z'' + z1]
  rm: runtime: ?, size: O(n^1) [z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 3*z + 8*z1

Computed RUNTIME bound using CoFloCo for: rm
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 9 + 8*z'

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

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

   eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0
   min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs)
   min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs)
   min(z) -{ 1 }-> 0 :|: z = 0
   min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   minsort(z) -{ 14 + 5*z + z^2 }-> 1 + s3 + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0
   minsort(z) -{ 68 + 16*x + 4*x*xs' + 2*x*y' + 2*x^2 + 23*xs' + 4*xs'*y' + 3*xs'^2 + 17*y' + 2*y'^2 }-> 1 + s4 + minsort(rm(s5, 1 + x + (1 + y' + xs'))) :|: s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   rm(z, z') -{ 5 + y2 }-> if2(s2, 1 + (z - 1), 1 + y2, xs) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0
   rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0
   rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0
   rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z]
  if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1]
  if2: runtime: O(n^1) [5 + 3*z + 8*z1], size: O(n^1) [1 + z'' + z1]
  rm: runtime: O(n^1) [9 + 8*z'], size: O(n^1) [z']

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

(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:

   eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 10 + 8*z1 }-> s17 :|: s17 >= 0, s17 <= z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 10 + 8*z1 }-> 1 + z'' + s18 :|: s18 >= 0, s18 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0
   min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs)
   min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs)
   min(z) -{ 1 }-> 0 :|: z = 0
   min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   minsort(z) -{ 23 + 13*z + z^2 }-> 1 + s3 + minsort(s11) :|: s11 >= 0, s11 <= 1 + (z - 1) + 0, s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0
   minsort(z) -{ 93 + 24*x + 4*x*xs' + 2*x*y' + 2*x^2 + 31*xs' + 4*xs'*y' + 3*xs'^2 + 25*y' + 2*y'^2 }-> 1 + s4 + minsort(s12) :|: s12 >= 0, s12 <= 1 + x + (1 + y' + xs'), s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   rm(z, z') -{ 2 + 8*z' }-> s13 :|: s13 >= 0, s13 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0
   rm(z, z') -{ 7 + 8*xs }-> s14 :|: s14 >= 0, s14 <= 1 + y1 + xs + 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0
   rm(z, z') -{ -1 + 8*z' }-> s15 :|: s15 >= 0, s15 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0
   rm(z, z') -{ 10 + 3*s2 + 8*xs + y2 }-> s16 :|: s16 >= 0, s16 <= 1 + y2 + xs + 1, s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0
   rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z]
  if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1]
  if2: runtime: O(n^1) [5 + 3*z + 8*z1], size: O(n^1) [1 + z'' + z1]
  rm: runtime: O(n^1) [9 + 8*z'], size: O(n^1) [z']

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

(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:

   eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 10 + 8*z1 }-> s17 :|: s17 >= 0, s17 <= z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 10 + 8*z1 }-> 1 + z'' + s18 :|: s18 >= 0, s18 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0
   min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs)
   min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs)
   min(z) -{ 1 }-> 0 :|: z = 0
   min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   minsort(z) -{ 23 + 13*z + z^2 }-> 1 + s3 + minsort(s11) :|: s11 >= 0, s11 <= 1 + (z - 1) + 0, s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0
   minsort(z) -{ 93 + 24*x + 4*x*xs' + 2*x*y' + 2*x^2 + 31*xs' + 4*xs'*y' + 3*xs'^2 + 25*y' + 2*y'^2 }-> 1 + s4 + minsort(s12) :|: s12 >= 0, s12 <= 1 + x + (1 + y' + xs'), s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   rm(z, z') -{ 2 + 8*z' }-> s13 :|: s13 >= 0, s13 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0
   rm(z, z') -{ 7 + 8*xs }-> s14 :|: s14 >= 0, s14 <= 1 + y1 + xs + 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0
   rm(z, z') -{ -1 + 8*z' }-> s15 :|: s15 >= 0, s15 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0
   rm(z, z') -{ 10 + 3*s2 + 8*xs + y2 }-> s16 :|: s16 >= 0, s16 <= 1 + y2 + xs + 1, s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0
   rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z]
  if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1]
  if2: runtime: O(n^1) [5 + 3*z + 8*z1], size: O(n^1) [1 + z'' + z1]
  rm: runtime: O(n^1) [9 + 8*z'], size: O(n^1) [z']
  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:

   eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 10 + 8*z1 }-> s17 :|: s17 >= 0, s17 <= z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'', z1) -{ 10 + 8*z1 }-> 1 + z'' + s18 :|: s18 >= 0, s18 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0
   min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs)
   min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs)
   min(z) -{ 1 }-> 0 :|: z = 0
   min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   minsort(z) -{ 23 + 13*z + z^2 }-> 1 + s3 + minsort(s11) :|: s11 >= 0, s11 <= 1 + (z - 1) + 0, s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0
   minsort(z) -{ 93 + 24*x + 4*x*xs' + 2*x*y' + 2*x^2 + 31*xs' + 4*xs'*y' + 3*xs'^2 + 25*y' + 2*y'^2 }-> 1 + s4 + minsort(s12) :|: s12 >= 0, s12 <= 1 + x + (1 + y' + xs'), s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   rm(z, z') -{ 2 + 8*z' }-> s13 :|: s13 >= 0, s13 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0
   rm(z, z') -{ 7 + 8*xs }-> s14 :|: s14 >= 0, s14 <= 1 + y1 + xs + 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0
   rm(z, z') -{ -1 + 8*z' }-> s15 :|: s15 >= 0, s15 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0
   rm(z, z') -{ 10 + 3*s2 + 8*xs + y2 }-> s16 :|: s16 >= 0, s16 <= 1 + y2 + xs + 1, s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0
   rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {minsort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z]
  if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1]
  if2: runtime: O(n^1) [5 + 3*z + 8*z1], size: O(n^1) [1 + z'' + z1]
  rm: runtime: O(n^1) [9 + 8*z'], size: O(n^1) [z']
  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)
   minsort(nil) -> nil
   minsort(cons(z0, z1)) -> cons(min(cons(z0, z1)), minsort(rm(min(cons(z0, z1)), cons(z0, z1))))
   min(nil) -> 0
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   rm(z0, nil) -> nil
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(z0, z2))
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(0, 0) -> c3
   EQ(0, s(z0)) -> c4
   EQ(s(z0), 0) -> c5
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(nil) -> c7
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1)))
   MIN(nil) -> c10
   MIN(cons(z0, nil)) -> c11
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, nil) -> c15
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(0, 0) -> c3
   EQ(0, s(z0)) -> c4
   EQ(s(z0), 0) -> c5
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(nil) -> c7
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1)))
   MIN(nil) -> c10
   MIN(cons(z0, nil)) -> c11
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, nil) -> c15
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
K tuples:none
Defined Rule Symbols:   le_2, eq_2, minsort_1, min_1, if1_4, rm_2, if2_4

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4

Compound Symbols:   c, c1, c2_1, c3, c4, c5, c6_1, c7, c8_1, c9_3, c10, c11, c12_2, c13_1, c14_1, c15, c16_2, c17_1, c18_1


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

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 9 trailing nodes:
   RM(z0, nil) -> c15
   MINSORT(nil) -> c7
   LE(s(z0), 0) -> c1
   EQ(0, 0) -> c3
   MIN(cons(z0, nil)) -> c11
   LE(0, z0) -> c
   EQ(0, s(z0)) -> c4
   MIN(nil) -> c10
   EQ(s(z0), 0) -> c5

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

(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)
   minsort(nil) -> nil
   minsort(cons(z0, z1)) -> cons(min(cons(z0, z1)), minsort(rm(min(cons(z0, z1)), cons(z0, z1))))
   min(nil) -> 0
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   rm(z0, nil) -> nil
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(z0, z2))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1)))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1)))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
K tuples:none
Defined Rule Symbols:   le_2, eq_2, minsort_1, min_1, if1_4, rm_2, if2_4

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4

Compound Symbols:   c2_1, c6_1, c8_1, c9_3, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1


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

(57) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   minsort(nil) -> nil
   minsort(cons(z0, z1)) -> cons(min(cons(z0, z1)), minsort(rm(min(cons(z0, z1)), cons(z0, z1))))
   min(nil) -> 0

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1)))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1)))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
K tuples:none
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4

Compound Symbols:   c2_1, c6_1, c8_1, c9_3, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1


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

(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)) -> c8(MIN(cons(z0, z1)))
We considered the (Usable) Rules:none
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1)))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   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_1
   POL(MINSORT(x_1)) = [1]
   POL(RM(x_1, x_2)) = 0
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(cons(x_1, x_2)) = 0
   POL(eq(x_1, x_2)) = [1] + x_1 + x_2
   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_2
   POL(le(x_1, x_2)) = [1] + x_1 + x_2
   POL(min(x_1)) = [1] + x_1
   POL(nil) = [1]
   POL(rm(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1
   POL(true) = [1]

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1)))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1)))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4

Compound Symbols:   c2_1, c6_1, c8_1, c9_3, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) by 
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(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))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4

Compound Symbols:   c2_1, c6_1, c8_1, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c9_3


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

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4

Compound Symbols:   c2_1, c6_1, c8_1, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c9_3, c9_2


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) by 
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2), LE(0, z0))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2), LE(s(z0), 0))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2), LE(0, z0))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2), LE(s(z0), 0))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2), LE(0, z0))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2), LE(s(z0), 0))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, RM_2, IF2_4, MIN_1

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c16_2, c17_1, c18_1, c9_3, c9_2, c12_2


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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, RM_2, IF2_4, MIN_1

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c16_2, c17_1, c18_1, c9_3, c9_2, c12_2, c12_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) by 
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2), EQ(0, 0))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2), EQ(0, s(z0)))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2), EQ(s(z0), 0))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2), EQ(0, 0))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2), EQ(0, s(z0)))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2), EQ(s(z0), 0))
   RM(s(z0), cons(s(z1), x2)) -> c16(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2), EQ(0, 0))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2), EQ(0, s(z0)))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2), EQ(s(z0), 0))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_3, c9_2, c12_2, c12_1, c16_2


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

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_3, c9_2, c12_2, c12_1, c16_2, c16_1


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) by 
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(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))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_3, c9_2, c12_2, c12_1, c16_2, c16_1


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

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_3, c9_2, c12_2, c12_1, c16_2, c16_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) by 
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_2, c12_2, c12_1, c16_2, c16_1, c9_3, c9_1


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

(79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(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))
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(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(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_1
   POL(MINSORT(x_1)) = [1]
   POL(RM(x_1, x_2)) = 0
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(cons(x_1, x_2)) = 0
   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)) = 0
   POL(nil) = 0
   POL(rm(x_1, x_2)) = [1] + x_2
   POL(s(x_1)) = 0
   POL(true) = 0

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_2, c12_2, c12_1, c16_2, c16_1, c9_3, c9_1


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) by 
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c12_2, c12_1, c16_2, c16_1, c9_3, c9_2, c9_1


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

(83) 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)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
We considered the (Usable) Rules:none
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 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_1
   POL(MINSORT(x_1)) = [1]
   POL(RM(x_1, x_2)) = 0
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(cons(x_1, x_2)) = 0
   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)) = 0
   POL(nil) = 0
   POL(rm(x_1, x_2)) = [1] + x_2
   POL(s(x_1)) = 0
   POL(true) = 0

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c12_2, c12_1, c16_2, c16_1, c9_3, c9_2, c9_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) by 
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1)))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c12_1, c16_2, c16_1, c9_3, c9_2, c9_1, c12_2


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) by 
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(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) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2


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

(89) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(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))
   IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2


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

(91) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) by 
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(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) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2

Compound Symbols:   c2_1, c6_1, c8_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1


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

(93) 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)))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(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))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0)))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2, LE_2

Compound Symbols:   c6_1, c8_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1, c2_1


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

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
S tuples:
   EQ(s(z0), s(z1)) -> c6(EQ(z0, z1))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2, LE_2

Compound Symbols:   c6_1, c8_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1, c2_1


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

(97) 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)))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
S tuples:
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0)))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0))))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2, LE_2, EQ_2

Compound Symbols:   c8_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1


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

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
S tuples:
   IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2, LE_2, EQ_2

Compound Symbols:   c8_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1


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

(101) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) by 
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
S tuples:
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, IF2_4, MIN_1, RM_2, IF1_4, LE_2, EQ_2

Compound Symbols:   c8_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1


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

(103) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
S tuples:
   IF2(true, z0, z1, z2) -> c17(RM(z0, z2))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, IF2_4, MIN_1, RM_2, IF1_4, LE_2, EQ_2

Compound Symbols:   c8_1, c17_1, c18_1, c12_1, c16_1, c9_2, c9_3, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1


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

(105) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) by 
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
S tuples:
   IF2(false, z0, z1, z2) -> c18(RM(z0, z2))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, IF2_4, MIN_1, RM_2, IF1_4, LE_2, EQ_2

Compound Symbols:   c8_1, c18_1, c12_1, c16_1, c9_2, c9_3, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1


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

(107) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) by 
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, MIN_1, RM_2, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c8_1, c12_1, c16_1, c9_2, c9_3, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1


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

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, MIN_1, RM_2, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c8_1, c12_1, c16_1, c9_2, c9_3, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1


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

(111) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2))))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, MIN_1, RM_2, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c8_1, c12_1, c16_1, c9_3, c9_1, c9_2, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1


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

(113) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2))))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, MIN_1, RM_2, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c8_1, c12_1, c16_1, c9_3, c9_1, c9_2, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1


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

(115) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) by MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1)))
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
K tuples:
   MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1)))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MINSORT_1, MIN_1, RM_2, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c8_1, c12_1, c16_1, c9_3, c9_1, c9_2, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1


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

(117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) by 
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2))))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c12_1, c16_1, c9_3, c9_1, c9_2, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c8_1


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

(119) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) by MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c12_1, c16_1, c9_3, c9_1, c9_2, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c8_1


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) by MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c12_1, c16_1, c9_1, c9_2, c12_2, c16_2, c9_3, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c8_1


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c12_1, c16_1, c9_2, c9_1, c12_2, c16_2, c9_3, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c8_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c12_1, c16_1, c9_1, c12_2, c16_2, c9_3, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c9_2, c8_1


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

(127) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) by MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c9_3, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c9_2, c8_1, c9_1


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

(129) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1)))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1


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

(131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) by 
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1)))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1


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

(133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) by 
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1)))
   IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1


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

(135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) by 
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
S tuples:
   MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2))
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1


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

(137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) by 
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(0), x1)))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1


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

(139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) by 
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(s(x0)), x2)))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1


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

(141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) by 
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, LE_2, EQ_2, IF1_4, MINSORT_1, IF2_4

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1, c13_1


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

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

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, EQ_2, IF1_4, MINSORT_1, IF2_4, LE_2

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1, c13_1, c2_1


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

(145) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_2, c9_3, c8_1, c9_1, c13_1, c2_1


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

(147) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_2, c9_3, c8_1, c9_1, c13_1, c2_1


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

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1


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

(151) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) by MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1


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

(153) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) by MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1


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

(155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) by 
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1)))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1


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

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

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1)))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1


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

(159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) by 
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) by MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
S tuples:
   MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2))
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   MIN_1, RM_2, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c12_1, c16_1, c12_2, c16_2, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1


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

(163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) by 
   MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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:
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
S tuples:
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2)))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   RM_2, MIN_1, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c16_1, c12_2, c16_2, c12_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1


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

(165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) by 
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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:
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))))
S tuples:
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1))))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   RM_2, MIN_1, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c16_1, c12_2, c16_2, c12_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1


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

(167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) by 
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, y3)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(s(y0))), cons(s(s(s(y1))), z2))) -> c12(IF1(le(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), LE(s(s(s(y0))), s(s(s(y1)))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(y3, cons(y4, y5)))), LE(s(s(z0)), s(s(z1))))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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:
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, y3)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(s(y0))), cons(s(s(s(y1))), z2))) -> c12(IF1(le(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), LE(s(s(s(y0))), s(s(s(y1)))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(y3, cons(y4, y5)))), LE(s(s(z0)), s(s(z1))))
S tuples:
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, y3)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(s(y0))), cons(s(s(s(y1))), z2))) -> c12(IF1(le(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), LE(s(s(s(y0))), s(s(s(y1)))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(y3, cons(y4, y5)))), LE(s(s(z0)), s(s(z1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   RM_2, MIN_1, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c16_1, c16_2, c12_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1, c12_2


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

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2)
   rm(z0, nil) -> nil
   min(cons(z0, nil)) -> z0
   min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> min(cons(z0, z2))
   if1(false, z0, z1, z2) -> min(cons(z1, z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if2(true, z0, z1, z2) -> rm(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, rm(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:
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(s(y0))), cons(s(s(s(y1))), z2))) -> c12(IF1(le(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), LE(s(s(s(y0))), s(s(s(y1)))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, y3)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c(LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c(LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c(LE(s(s(z0)), s(s(z1))))
S tuples:
   RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2))
   RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2))
   RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2))
   RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2))
   RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2))
   RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2))
   RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2))
   IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1)))
   IF2(true, 0, 0, x0) -> c17(RM(0, x0))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2))
   IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0))
   IF2(false, 0, s(x0), x1) -> c18(RM(0, x1))
   IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1))
   IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2))
   IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1))
   IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1))
   MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2))))
   MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil)))
   MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1))))
   RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1))))
   IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1))))
   MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2)))
   IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1))))
   IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2))))
   IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1))))
   MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1))))
   MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1)))
   MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1))))
   IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2)))))
   MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2))))
   MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2))))
   IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1))))
   IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(s(y0))), cons(s(s(s(y1))), z2))) -> c12(IF1(le(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), LE(s(s(s(y0))), s(s(s(y1)))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, y3)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c(LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c(LE(s(s(z0)), s(s(z1))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))
   MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c(LE(s(s(z0)), s(s(z1))))
K tuples:
   MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil)))
   MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1))))
   MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1))))
   MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2))))
   MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2))))
   MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1))))
   MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1))))
Defined Rule Symbols:   rm_2, min_1, if1_4, le_2, if2_4, eq_2

Defined Pair Symbols:   RM_2, MIN_1, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2

Compound Symbols:   c16_1, c16_2, c12_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1, c12_2, c_1