WORST_CASE(?,O(n^2))
proof of input_egMfbIKvYK.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 216 ms]
(2) CpxRelTRS
(3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
(5) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(6) CdtProblem
(7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 664 ms]
(10) CdtProblem
(11) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 376 ms]
(14) CdtProblem
(15) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CdtProblem
(19) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CdtProblem
(21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 604 ms]
(22) CdtProblem
(23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 382 ms]
(24) CdtProblem
(25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CdtProblem
(27) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(28) CdtProblem
(29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(30) CdtProblem
(31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(32) CdtProblem
(33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(34) CdtProblem
(35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CdtProblem
(37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(38) CdtProblem
(39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(40) CdtProblem
(41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2437 ms]
(42) CdtProblem
(43) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 3300 ms]
(44) CdtProblem
(45) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 4562 ms]
(46) CdtProblem
(47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
(49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(50) CdtProblem
(51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(52) CdtProblem
(53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(54) CdtProblem
(55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(56) CdtProblem
(57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(58) CdtProblem
(59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(60) CdtProblem
(61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(62) CdtProblem
(63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(64) CdtProblem
(65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(66) CdtProblem
(67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
(68) CdtProblem
(69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(70) CdtProblem
(71) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(72) CdtProblem
(73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(74) CdtProblem
(75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(76) CdtProblem
(77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(78) CdtProblem
(79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(80) CdtProblem
(81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(82) CdtProblem
(83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(84) CdtProblem
(85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(86) CdtProblem
(87) CdtLeafRemovalProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
(94) CdtProblem
(95) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(96) CdtProblem
(97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(98) CdtProblem
(99) CdtLeafRemovalProof [ComplexityIfPolyImplication, 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) CdtRewritingProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(114) CdtProblem
(115) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(116) CdtProblem
(117) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(118) CdtProblem
(119) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(120) CdtProblem
(121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
(122) CdtProblem
(123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(124) CdtProblem
(125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(126) CdtProblem
(127) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 5376 ms]
(128) CdtProblem
(129) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(130) BOUNDS(1, 1)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
   mergesort(Cons(x, Nil)) -> Cons(x, Nil)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
   splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
   mergesort(Nil) -> Nil
   merge(Nil, xs2) -> xs2
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> mergesort(xs)

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

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
   merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

   mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
   mergesort(Cons(x, Nil)) -> Cons(x, Nil)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
   splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
   mergesort(Nil) -> Nil
   merge(Nil, xs2) -> xs2
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> mergesort(xs)

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

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
   merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> mergesort(z0)
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   <='(0, z0) -> c1
   <='(S(z0), 0) -> c2
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGESORT(Cons(z0, Nil)) -> c6
   MERGESORT(Nil) -> c7
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   NOTEMPTY(Cons(z0, z1)) -> c14
   NOTEMPTY(Nil) -> c15
   GOAL(z0) -> c16(MERGESORT(z0))
S tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGESORT(Cons(z0, Nil)) -> c6
   MERGESORT(Nil) -> c7
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   NOTEMPTY(Cons(z0, z1)) -> c14
   NOTEMPTY(Nil) -> c15
   GOAL(z0) -> c16(MERGESORT(z0))
K tuples:none
Defined Rule Symbols:   mergesort_1, merge_2, splitmerge_3, notEmpty_1, goal_1, <=_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3, NOTEMPTY_1, GOAL_1

Compound Symbols:   c_1, c1, c2, c3_1, c4_1, c5_1, c6, c7, c8_2, c9, c10, c11_1, c12_2, c13_2, c14, c15, c16_1


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

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0) -> c16(MERGESORT(z0))
Removed 6 trailing nodes:
   MERGESORT(Nil) -> c7
   <='(0, z0) -> c1
   NOTEMPTY(Cons(z0, z1)) -> c14
   NOTEMPTY(Nil) -> c15
   MERGESORT(Cons(z0, Nil)) -> c6
   <='(S(z0), 0) -> c2

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> mergesort(z0)
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
S tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
K tuples:none
Defined Rule Symbols:   mergesort_1, merge_2, splitmerge_3, notEmpty_1, goal_1, <=_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_2, c13_2


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> mergesort(z0)

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
S tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
K tuples:none
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_2, c13_2


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
We considered the (Usable) Rules:none
And the Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<=(x_1, x_2)) = 0
   POL(<='(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = 0
   POL(MERGE(x_1, x_2)) = 0
   POL(MERGESORT(x_1)) = [2]x_1^2
   POL(MERGE[ITE](x_1, x_2, x_3)) = 0
   POL(Nil) = 0
   POL(S(x_1)) = 0
   POL(SPLITMERGE(x_1, x_2, x_3)) = [2]x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2
   POL(True) = 0
   POL(c(x_1)) = x_1
   POL(c10) = 0
   POL(c11(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9) = 0
   POL(merge(x_1, x_2)) = [1] + x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2
   POL(merge[Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 + x_3^2 + x_2*x_3 + x_2^2
   POL(mergesort(x_1)) = 0
   POL(splitmerge(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 + x_3^2 + x_2*x_3 + x_1*x_3 + x_1^2 + x_1*x_2 + x_2^2

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
S tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_2, c13_2


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

(11) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
S tuples:
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_2, c13_2


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

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
We considered the (Usable) Rules:none
And the Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<=(x_1, x_2)) = 0
   POL(<='(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = 0
   POL(MERGE(x_1, x_2)) = [1]
   POL(MERGESORT(x_1)) = [2]x_1^2
   POL(MERGE[ITE](x_1, x_2, x_3)) = [1]
   POL(Nil) = 0
   POL(S(x_1)) = 0
   POL(SPLITMERGE(x_1, x_2, x_3)) = [1] + x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2
   POL(True) = 0
   POL(c(x_1)) = x_1
   POL(c10) = 0
   POL(c11(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9) = 0
   POL(merge(x_1, x_2)) = [1] + x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2
   POL(merge[Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 + x_3^2 + x_2*x_3 + x_2^2
   POL(mergesort(x_1)) = 0
   POL(splitmerge(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 + x_3^2 + x_2*x_3 + x_1*x_3 + x_1^2 + x_1*x_2 + x_2^2

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
S tuples:
   MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_2, c13_2


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

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) by 
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(z0, z1), Nil) -> c9
   MERGE(Nil, z0) -> c10
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c9, c10, c11_1, c12_2, c13_2, c8_2


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

(17) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   MERGE(Nil, z0) -> c10
   MERGE(Cons(z0, z1), Nil) -> c9

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c12_2, c13_2, c8_2


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

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c12_2, c13_2, c8_2, c8_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
We considered the (Usable) Rules:
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   <=(S(z0), 0) -> False
   merge(Nil, z0) -> z0
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Nil) -> Nil
   <=(0, z0) -> True
   <=(S(z0), S(z1)) -> <=(z0, z1)
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
And the Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<=(x_1, x_2)) = [1]
   POL(<='(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = 0
   POL(MERGE(x_1, x_2)) = [1] + [2]x_2
   POL(MERGESORT(x_1)) = [2]x_1^2
   POL(MERGE[ITE](x_1, x_2, x_3)) = [2]x_3 + x_1^2
   POL(Nil) = 0
   POL(S(x_1)) = 0
   POL(SPLITMERGE(x_1, x_2, x_3)) = [1] + x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2
   POL(True) = [1]
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(merge(x_1, x_2)) = x_1 + x_2
   POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3
   POL(mergesort(x_1)) = x_1
   POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c12_2, c13_2, c8_2, c8_1


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

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
We considered the (Usable) Rules:
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   <=(S(z0), 0) -> False
   merge(Nil, z0) -> z0
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Nil) -> Nil
   <=(0, z0) -> True
   <=(S(z0), S(z1)) -> <=(z0, z1)
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
And the Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<=(x_1, x_2)) = [2]
   POL(<='(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [2] + x_2
   POL(False) = [2]
   POL(MERGE(x_1, x_2)) = [2]x_2 + x_1*x_2
   POL(MERGESORT(x_1)) = x_1^2
   POL(MERGE[ITE](x_1, x_2, x_3)) = x_2*x_3 + x_1*x_3
   POL(Nil) = 0
   POL(S(x_1)) = 0
   POL(SPLITMERGE(x_1, x_2, x_3)) = [2]x_3 + x_3^2 + x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2
   POL(True) = 0
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(merge(x_1, x_2)) = x_1 + x_2
   POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3
   POL(mergesort(x_1)) = x_1
   POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c12_2, c13_2, c8_2, c8_1


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

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0)) by 
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGE(mergesort(x0), Nil), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Nil, x1) -> c12(MERGE(Nil, mergesort(x1)), MERGESORT(Nil))

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGE(mergesort(x0), Nil), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Nil, x1) -> c12(MERGE(Nil, mergesort(x1)), MERGESORT(Nil))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c13_2, c8_2, c8_1, c12_2


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

(27) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SPLITMERGE(Nil, Nil, x1) -> c12(MERGE(Nil, mergesort(x1)), MERGESORT(Nil))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGE(mergesort(x0), Nil), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(Cons(z0, Nil)))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c13_2, c8_2, c8_1, c12_2


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

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c13_2, c8_2, c8_1, c12_2, c12_1


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

(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1)) by 
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, x0, Nil) -> c13(MERGE(mergesort(x0), Nil), MERGESORT(Nil))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Nil, x1) -> c13(MERGE(Nil, mergesort(x1)), MERGESORT(x1))

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, x0, Nil) -> c13(MERGE(mergesort(x0), Nil), MERGESORT(Nil))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Nil, x1) -> c13(MERGE(Nil, mergesort(x1)), MERGESORT(x1))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   SPLITMERGE(Nil, z0, z1) -> c13(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_2, c12_1, c13_2


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

(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SPLITMERGE(Nil, x0, Nil) -> c13(MERGE(mergesort(x0), Nil), MERGESORT(Nil))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Nil, x1) -> c13(MERGE(Nil, mergesort(x1)), MERGESORT(x1))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_2, c12_1, c13_2


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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   SPLITMERGE(Nil, Nil, x1) -> c13(MERGESORT(x1))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_2, c12_1, c13_2, c13_1


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

(37) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   SPLITMERGE(Nil, Nil, x1) -> c13(MERGESORT(x1))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
S tuples:
   MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_2, c12_1, c13_2, c13_1


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

(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) by 
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2


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

(41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
We considered the (Usable) Rules:
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Nil) -> Nil
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   merge(Nil, z0) -> z0
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
And the Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<=(x_1, x_2)) = 0
   POL(<='(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [2] + x_2
   POL(False) = 0
   POL(MERGE(x_1, x_2)) = x_1*x_2
   POL(MERGESORT(x_1)) = x_1^2
   POL(MERGE[ITE](x_1, x_2, x_3)) = x_2*x_3
   POL(Nil) = 0
   POL(S(x_1)) = 0
   POL(SPLITMERGE(x_1, x_2, x_3)) = x_3^2 + x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2
   POL(True) = 0
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(merge(x_1, x_2)) = x_1 + x_2
   POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3
   POL(mergesort(x_1)) = x_1
   POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2


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

(43) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
We considered the (Usable) Rules:
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   <=(S(z0), 0) -> False
   merge(Nil, z0) -> z0
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Nil) -> Nil
   <=(0, z0) -> True
   <=(S(z0), S(z1)) -> <=(z0, z1)
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
And the Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<=(x_1, x_2)) = [1]
   POL(<='(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [2] + x_2
   POL(False) = [1]
   POL(MERGE(x_1, x_2)) = x_2 + x_1*x_2
   POL(MERGESORT(x_1)) = x_1^2
   POL(MERGE[ITE](x_1, x_2, x_3)) = x_2*x_3 + x_1*x_3
   POL(Nil) = 0
   POL(S(x_1)) = 0
   POL(SPLITMERGE(x_1, x_2, x_3)) = x_3 + x_3^2 + x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2
   POL(True) = 0
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(merge(x_1, x_2)) = x_1 + x_2
   POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3
   POL(mergesort(x_1)) = x_1
   POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2


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

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
We considered the (Usable) Rules:
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   <=(S(z0), 0) -> False
   merge(Nil, z0) -> z0
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Nil) -> Nil
   <=(0, z0) -> True
   <=(S(z0), S(z1)) -> <=(z0, z1)
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
And the Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<=(x_1, x_2)) = [1]
   POL(<='(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = 0
   POL(MERGE(x_1, x_2)) = [2] + [2]x_2
   POL(MERGESORT(x_1)) = [2]x_1^2
   POL(MERGE[ITE](x_1, x_2, x_3)) = [2]x_3 + [2]x_1^2
   POL(Nil) = 0
   POL(S(x_1)) = 0
   POL(SPLITMERGE(x_1, x_2, x_3)) = [2] + x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2
   POL(True) = [1]
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(merge(x_1, x_2)) = x_1 + x_2
   POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3
   POL(mergesort(x_1)) = x_1
   POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0)) by 
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c12(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Nil))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c12(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Nil))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2


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

(49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c12(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Nil))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Nil)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2


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

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) by 
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c12(MERGE(Nil, Cons(x1, Nil)), MERGESORT(Nil))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c12(MERGE(Nil, Cons(x1, Nil)), MERGESORT(Nil))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2


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

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c12(MERGE(Nil, Cons(x1, Nil)), MERGESORT(Nil))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Nil)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2


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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c13_2, c13_1, c8_2


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2)))) by 
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2


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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1))) by 
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Cons(z1, z2))) -> c12(MERGE(Cons(x0, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c12(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c12(MERGE(Cons(x0, Nil), Nil))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c12(MERGE(Cons(x0, Nil), Nil))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2


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

(65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c12(MERGE(Cons(x0, Nil), Nil))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by 
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c13(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c13(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2


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

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2


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

(71) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(x1)) by 
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Nil))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Nil))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2


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

(75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Nil))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_2, c13_1, c8_2, c12_2


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c13(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) by 
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Cons(z1, z2))) -> c13(MERGE(Cons(x0, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c13(MERGE(Cons(x0, Nil), Nil), MERGESORT(Nil))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c13(MERGE(Cons(x0, Nil), Nil), MERGESORT(Nil))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_1, c8_2, c12_2, c13_2


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

(81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c13(MERGE(Cons(x0, Nil), Nil), MERGESORT(Nil))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_1, c8_2, c12_2, c13_2


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

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c13_1, c8_2, c12_2, c13_2


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c13(MERGE(mergesort(x0), Cons(z0, Nil))) by 
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c13(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c13(MERGE(Nil, Cons(x1, Nil)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c13(MERGE(Nil, Cons(x1, Nil)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1


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

(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c13(MERGE(Nil, Cons(x1, Nil)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1


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

(89) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0)) by SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1


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

(91) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) by 
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c3_1


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

(93) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2

Compound Symbols:   c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c3_1


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

(95) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) by 
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, SPLITMERGE_3, MERGE_2, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c3_1, c4_1


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

(97) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) by 
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c12_1, c8_2, c12_2, c13_2, c13_1, c3_1, c4_1, c11_1


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

(99) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c12_2, c13_2, c13_1, c3_1, c4_1, c11_1


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

(101) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2)))) by 
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c12_2, c13_2, c13_1, c3_1, c4_1, c11_1


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

(103) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c12_2, c13_2, c13_1, c3_1, c4_1, c11_1


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

(105) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c12_2, c13_2, c13_1, c3_1, c4_1, c11_1


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

(107) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) by 
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c12_2, c13_2, c13_1, c3_1, c4_1, c11_1


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

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c12_2, c13_2, c13_1, c3_1, c4_1, c11_1


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

(111) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c3_1, c4_1, c11_1


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

(113) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c13(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(Cons(z0, Cons(x2, x3)))) by 
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3))))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c3_1, c4_1, c11_1


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

(115) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c13(MERGESORT(Cons(x1, Cons(x2, x3)))) by 
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3))))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c3_1, c4_1, c11_1


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

(117) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c13(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(x3)) by 
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c12_2, c3_1, c4_1, c11_1


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

(119) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0)) by 
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3)))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c3_1, c12_2, c4_1, c11_1


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Cons(z1, z2))) -> c12(MERGE(Cons(x0, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   <='(S(z0), S(z1)) -> c(<='(z0, z1))
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3)))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   <='_2, MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3

Compound Symbols:   c_1, c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c3_1, c12_2, c4_1, c11_1


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

(123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace <='(S(z0), S(z1)) -> c(<='(z0, z1)) by 
   <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1)))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3)))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)))
   <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2

Compound Symbols:   c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c3_1, c12_2, c4_1, c11_1, c_1


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

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3)))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)))
   <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)))
S tuples:
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2

Compound Symbols:   c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c3_1, c12_2, c4_1, c11_1, c_1


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

(127) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
We considered the (Usable) Rules:
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Nil) -> Nil
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   merge(Nil, z0) -> z0
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
And the Tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3)))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)))
   <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<=(x_1, x_2)) = 0
   POL(<='(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [2] + x_2
   POL(False) = 0
   POL(MERGE(x_1, x_2)) = [1] + x_1 + x_2
   POL(MERGESORT(x_1)) = [1] + x_1^2
   POL(MERGE[ITE](x_1, x_2, x_3)) = x_2 + x_3
   POL(Nil) = 0
   POL(S(x_1)) = 0
   POL(SPLITMERGE(x_1, x_2, x_3)) = x_3^2 + [2]x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2
   POL(True) = 0
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(merge(x_1, x_2)) = x_1 + x_2
   POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3
   POL(mergesort(x_1)) = [2]x_1
   POL(splitmerge(x_1, x_2, x_3)) = [2]x_1 + [2]x_2 + [2]x_3

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <=(S(z0), S(z1)) -> <=(z0, z1)
   <=(0, z0) -> True
   <=(S(z0), 0) -> False
   mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)
   mergesort(Cons(z0, Nil)) -> Cons(z0, Nil)
   mergesort(Nil) -> Nil
   splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2)
   splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1))
   merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3))
   merge(Cons(z0, z1), Nil) -> Cons(z0, z1)
   merge(Nil, z0) -> z0
   merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2))
   merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2))
Tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c13(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c13(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3))))
   SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c13(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)))
   SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c13(MERGE(Cons(x0, Nil), Cons(z0, Nil)))
   MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3))
   MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2)))
   MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3)))
   MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)))
   SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGE(mergesort(Cons(x0, x4)), splitmerge(Cons(z2, z3), Cons(x2, Nil), Nil)), MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c13(MERGESORT(Cons(x2, Cons(z2, z3))))
   SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c13(MERGE(splitmerge(Cons(z1, z2), Cons(x0, Nil), Nil), mergesort(Cons(x2, x3))), MERGESORT(Cons(x2, x3)))
   SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)))
   <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)))
S tuples:none
K tuples:
   MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil))
   MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)))
   MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)))
   MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2)))
   SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil))
   SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3)))
   MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)))
   MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1))))
Defined Rule Symbols:   <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3

Defined Pair Symbols:   MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2

Compound Symbols:   c5_1, c8_1, c8_2, c12_1, c13_2, c13_1, c3_1, c12_2, c4_1, c11_1, c_1


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(129) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(130)
BOUNDS(1, 1)