YES(O(1), O(n^2)) 626.37/230.56 YES(O(1), O(n^2)) 630.77/231.58 630.77/231.58 630.77/231.58 630.77/231.58 630.77/231.58 630.77/231.58 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 630.77/231.58 630.77/231.58 630.77/231.58
630.77/231.58
630.77/231.58
630.77/231.58
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
630.77/231.58 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳2 CdtProblem
630.77/231.58 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳4 CdtProblem
630.77/231.58 
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳6 CdtProblem
630.77/231.58 
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳8 CdtProblem
630.77/231.58 
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳10 CdtProblem
630.77/231.58 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
630.77/231.58 
↳12 CdtProblem
630.77/231.58 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
630.77/231.58 
↳14 CdtProblem
630.77/231.58 
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
630.77/231.58 
↳16 CdtProblem
630.77/231.58 
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳18 CdtProblem
630.77/231.58 
↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳20 CdtProblem
630.77/231.58 
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳22 CdtProblem
630.77/231.58 
↳23 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳24 CdtProblem
630.77/231.58 
↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
630.77/231.58 
↳26 CdtProblem
630.77/231.58 
↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
630.77/231.58 
↳28 CdtProblem
630.77/231.58 
↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳30 CdtProblem
630.77/231.58 
↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
630.77/231.58 
↳32 CdtProblem
630.77/231.58 
↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
630.77/231.58 
↳34 CdtProblem
630.77/231.58 
↳35 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
630.77/231.58 
↳36 CdtProblem
630.77/231.58 
↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳38 CdtProblem
630.77/231.58 
↳39 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳40 CdtProblem
630.77/231.58 
↳41 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳42 CdtProblem
630.77/231.58 
↳43 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳44 CdtProblem
630.77/231.58 
↳45 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳46 CdtProblem
630.77/231.58 
↳47 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳48 CdtProblem
630.77/231.58 
↳49 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳50 CdtProblem
630.77/231.58 
↳51 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳52 CdtProblem
630.77/231.58 
↳53 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳54 CdtProblem
630.77/231.58 
↳55 CdtRewritingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳56 CdtProblem
630.77/231.58 
↳57 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳58 CdtProblem
630.77/231.58 
↳59 CdtRewritingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳60 CdtProblem
630.77/231.58 
↳61 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳62 CdtProblem
630.77/231.58 
↳63 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳64 CdtProblem
630.77/231.58 
↳65 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳66 CdtProblem
630.77/231.58 
↳67 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳68 CdtProblem
630.77/231.58 
↳69 CdtRewritingProof (BOTH BOUNDS(ID, ID))
630.77/231.58 
↳70 CdtProblem
630.77/231.58 
↳71 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳72 CdtProblem
631.19/231.65 
↳73 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
631.19/231.65 
↳74 CdtProblem
631.19/231.65 
↳75 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳76 CdtProblem
631.19/231.65 
↳77 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳78 CdtProblem
631.19/231.65 
↳79 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳80 CdtProblem
631.19/231.65 
↳81 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳82 CdtProblem
631.19/231.65 
↳83 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳84 CdtProblem
631.19/231.65 
↳85 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳86 CdtProblem
631.19/231.65 
↳87 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳88 CdtProblem
631.19/231.65 
↳89 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳90 CdtProblem
631.19/231.65 
↳91 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳92 CdtProblem
631.19/231.65 
↳93 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳94 CdtProblem
631.19/231.65 
↳95 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳96 CdtProblem
631.19/231.65 
↳97 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
631.19/231.65 
↳98 CdtProblem
631.19/231.65 
↳99 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳100 CdtProblem
631.19/231.65 
↳101 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳102 CdtProblem
631.19/231.65 
↳103 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳104 CdtProblem
631.19/231.65 
↳105 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳106 CdtProblem
631.19/231.65 
↳107 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
631.19/231.65 
↳108 CdtProblem
631.19/231.65 
↳109 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳110 CdtProblem
631.19/231.65 
↳111 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳112 CdtProblem
631.19/231.65 
↳113 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳114 CdtProblem
631.19/231.65 
↳115 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳116 CdtProblem
631.19/231.65 
↳117 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳118 CdtProblem
631.19/231.65 
↳119 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳120 CdtProblem
631.19/231.65 
↳121 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳122 CdtProblem
631.19/231.65 
↳123 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳124 CdtProblem
631.19/231.65 
↳125 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳126 CdtProblem
631.19/231.65 
↳127 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳128 CdtProblem
631.19/231.65 
↳129 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳130 CdtProblem
631.19/231.65 
↳131 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳132 CdtProblem
631.19/231.65 
↳133 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳134 CdtProblem
631.19/231.65 
↳135 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳136 CdtProblem
631.19/231.65 
↳137 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳138 CdtProblem
631.19/231.65 
↳139 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳140 CdtProblem
631.19/231.65 
↳141 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳142 CdtProblem
631.19/231.65 
↳143 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳144 CdtProblem
631.19/231.65 
↳145 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳146 CdtProblem
631.19/231.65 
↳147 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳148 CdtProblem
631.19/231.65 
↳149 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳150 CdtProblem
631.19/231.65 
↳151 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳152 CdtProblem
631.19/231.65 
↳153 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳154 CdtProblem
631.19/231.65 
↳155 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳156 CdtProblem
631.19/231.65 
↳157 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳158 CdtProblem
631.19/231.65 
↳159 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳160 CdtProblem
631.19/231.65 
↳161 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳162 CdtProblem
631.19/231.65 
↳163 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳164 CdtProblem
631.19/231.65 
↳165 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳166 CdtProblem
631.19/231.65 
↳167 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳168 CdtProblem
631.19/231.65 
↳169 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳170 CdtProblem
631.19/231.65 
↳171 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳172 CdtProblem
631.19/231.65 
↳173 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳174 CdtProblem
631.19/231.65 
↳175 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳176 CdtProblem
631.19/231.65 
↳177 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳178 CdtProblem
631.19/231.65 
↳179 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳180 CdtProblem
631.19/231.65 
↳181 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳182 CdtProblem
631.19/231.65 
↳183 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳184 CdtProblem
631.19/231.65 
↳185 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳186 CdtProblem
631.19/231.65 
↳187 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳188 CdtProblem
631.19/231.65 
↳189 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳190 CdtProblem
631.19/231.65 
↳191 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳192 CdtProblem
631.19/231.65 
↳193 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳194 CdtProblem
631.19/231.65 
↳195 CdtRewritingProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳196 CdtProblem
631.19/231.65 
↳197 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳198 CdtProblem
631.19/231.65 
↳199 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
631.19/231.65 
↳200 CdtProblem
631.19/231.65 
↳201 SIsEmptyProof (BOTH BOUNDS(ID, ID))
631.19/231.65 
↳202 BOUNDS(O(1), O(1))
631.19/231.65
631.19/231.65 631.19/231.65
631.19/231.65
631.19/231.65

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

cond(true, x) → cond(and(even(x), gr(x, 0)), p(x)) 631.19/231.65
and(x, false) → false 631.19/231.65
and(false, x) → false 631.19/231.65
and(true, true) → true 631.19/231.65
even(0) → true 631.19/231.65
even(s(0)) → false 631.19/231.65
even(s(s(x))) → even(x) 631.19/231.65
gr(0, x) → false 631.19/231.65
gr(s(x), 0) → true 631.19/231.65
gr(s(x), s(y)) → gr(x, y) 631.19/231.65
p(0) → 0 631.19/231.65
p(s(x)) → x

Rewrite Strategy: INNERMOST
631.19/231.65
631.19/231.65

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
631.19/231.65
631.19/231.65

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.19/231.65
and(z0, false) → false 631.19/231.65
and(false, z0) → false 631.19/231.65
and(true, true) → true 631.19/231.65
even(0) → true 631.19/231.65
even(s(0)) → false 631.19/231.65
even(s(s(z0))) → even(z0) 631.19/231.65
gr(0, z0) → false 631.19/231.65
gr(s(z0), 0) → true 631.19/231.65
gr(s(z0), s(y)) → gr(z0, y) 631.19/231.65
p(0) → 0 631.19/231.65
p(s(z0)) → z0
Tuples:

COND(true, z0) → c(COND(and(even(z0), gr(z0, 0)), p(z0)), AND(even(z0), gr(z0, 0)), EVEN(z0), GR(z0, 0), P(z0)) 631.19/231.65
EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.65
GR(s(z0), s(y)) → c9(GR(z0, y))
S tuples:

COND(true, z0) → c(COND(and(even(z0), gr(z0, 0)), p(z0)), AND(even(z0), gr(z0, 0)), EVEN(z0), GR(z0, 0), P(z0)) 631.19/231.65
EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.65
GR(s(z0), s(y)) → c9(GR(z0, y))
K tuples:none
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN, GR

Compound Symbols:

c, c6, c9

631.19/231.65
631.19/231.65

(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts
631.19/231.65
631.19/231.65

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.19/231.65
and(z0, false) → false 631.19/231.65
and(false, z0) → false 631.19/231.65
and(true, true) → true 631.19/231.65
even(0) → true 631.19/231.65
even(s(0)) → false 631.19/231.65
even(s(s(z0))) → even(z0) 631.19/231.65
gr(0, z0) → false 631.19/231.65
gr(s(z0), 0) → true 631.19/231.65
gr(s(z0), s(y)) → gr(z0, y) 631.19/231.65
p(0) → 0 631.19/231.65
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.65
COND(true, z0) → c(COND(and(even(z0), gr(z0, 0)), p(z0)), EVEN(z0)) 631.19/231.65
GR(s(z0), s(y)) → c9
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.65
COND(true, z0) → c(COND(and(even(z0), gr(z0, 0)), p(z0)), EVEN(z0)) 631.19/231.65
GR(s(z0), s(y)) → c9
K tuples:none
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND, GR

Compound Symbols:

c6, c, c9

631.19/231.65
631.19/231.65

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

GR(s(z0), s(y)) → c9
631.19/231.65
631.19/231.65

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.19/231.65
and(z0, false) → false 631.19/231.65
and(false, z0) → false 631.19/231.65
and(true, true) → true 631.19/231.65
even(0) → true 631.19/231.65
even(s(0)) → false 631.19/231.65
even(s(s(z0))) → even(z0) 631.19/231.65
gr(0, z0) → false 631.19/231.65
gr(s(z0), 0) → true 631.19/231.65
gr(s(z0), s(y)) → gr(z0, y) 631.19/231.65
p(0) → 0 631.19/231.65
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.65
COND(true, z0) → c(COND(and(even(z0), gr(z0, 0)), p(z0)), EVEN(z0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.65
COND(true, z0) → c(COND(and(even(z0), gr(z0, 0)), p(z0)), EVEN(z0))
K tuples:none
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c

631.19/231.65
631.19/231.65

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, z0) → c(COND(and(even(z0), gr(z0, 0)), p(z0)), EVEN(z0)) by

COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0), EVEN(0)) 631.19/231.65
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.65
COND(true, 0) → c(COND(and(even(0), false), p(0)), EVEN(0)) 631.19/231.65
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.65
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0)), EVEN(0)) 631.19/231.65
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))), EVEN(s(0))) 631.19/231.65
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0))))
631.19/231.65
631.19/231.65

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.19/231.65
and(z0, false) → false 631.19/231.65
and(false, z0) → false 631.19/231.65
and(true, true) → true 631.19/231.65
even(0) → true 631.19/231.65
even(s(0)) → false 631.19/231.65
even(s(s(z0))) → even(z0) 631.19/231.65
gr(0, z0) → false 631.19/231.65
gr(s(z0), 0) → true 631.19/231.65
gr(s(z0), s(y)) → gr(z0, y) 631.19/231.65
p(0) → 0 631.19/231.65
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.65
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0), EVEN(0)) 631.19/231.65
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.65
COND(true, 0) → c(COND(and(even(0), false), p(0)), EVEN(0)) 631.19/231.65
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.65
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0)), EVEN(0)) 631.19/231.65
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))), EVEN(s(0))) 631.19/231.65
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.65
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0), EVEN(0)) 631.19/231.65
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.65
COND(true, 0) → c(COND(and(even(0), false), p(0)), EVEN(0)) 631.19/231.65
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.65
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0)), EVEN(0)) 631.19/231.65
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))), EVEN(s(0))) 631.19/231.65
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0))))
K tuples:none
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c

631.19/231.65
631.19/231.65

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts
631.19/231.65
631.19/231.65

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.19/231.65
and(z0, false) → false 631.19/231.65
and(false, z0) → false 631.19/231.65
and(true, true) → true 631.19/231.65
even(0) → true 631.19/231.65
even(s(0)) → false 631.19/231.65
even(s(s(z0))) → even(z0) 631.19/231.65
gr(0, z0) → false 631.19/231.65
gr(s(z0), 0) → true 631.19/231.65
gr(s(z0), s(y)) → gr(z0, y) 631.19/231.65
p(0) → 0 631.19/231.65
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.65
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))))
K tuples:none
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.19/231.69
631.19/231.69

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))))
We considered the (Usable) Rules:

gr(s(z0), 0) → true 631.19/231.69
gr(0, z0) → false 631.19/231.69
and(z0, false) → false 631.19/231.69
and(false, z0) → false 631.19/231.69
and(true, true) → true 631.19/231.69
p(s(z0)) → z0 631.19/231.69
p(0) → 0 631.19/231.69
even(0) → true 631.19/231.69
even(s(0)) → false 631.19/231.69
even(s(s(z0))) → even(z0)
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.19/231.69

POL(0) = 0    631.19/231.69
POL(COND(x1, x2)) = [4]x1    631.19/231.69
POL(EVEN(x1)) = 0    631.19/231.69
POL(and(x1, x2)) = x1    631.19/231.69
POL(c(x1)) = x1    631.19/231.69
POL(c(x1, x2)) = x1 + x2    631.19/231.69
POL(c6(x1)) = x1    631.19/231.69
POL(even(x1)) = [4]    631.19/231.69
POL(false) = 0    631.19/231.69
POL(gr(x1, x2)) = [2]x1    631.19/231.69
POL(p(x1)) = 0    631.19/231.69
POL(s(x1)) = [4]    631.19/231.69
POL(true) = [4]   
631.19/231.69
631.19/231.69

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.19/231.69
and(z0, false) → false 631.19/231.69
and(false, z0) → false 631.19/231.69
and(true, true) → true 631.19/231.69
even(0) → true 631.19/231.69
even(s(0)) → false 631.19/231.69
even(s(s(z0))) → even(z0) 631.19/231.69
gr(0, z0) → false 631.19/231.69
gr(s(z0), 0) → true 631.19/231.69
gr(s(z0), s(y)) → gr(z0, y) 631.19/231.69
p(0) → 0 631.19/231.69
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0)))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.19/231.69
631.19/231.69

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0)))
We considered the (Usable) Rules:

gr(s(z0), 0) → true 631.19/231.69
gr(0, z0) → false 631.19/231.69
and(z0, false) → false 631.19/231.69
and(false, z0) → false 631.19/231.69
and(true, true) → true 631.19/231.69
p(s(z0)) → z0 631.19/231.69
p(0) → 0 631.19/231.69
even(0) → true 631.19/231.69
even(s(0)) → false 631.19/231.69
even(s(s(z0))) → even(z0)
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.19/231.69

POL(0) = 0    631.19/231.69
POL(COND(x1, x2)) = [2]x1    631.19/231.69
POL(EVEN(x1)) = 0    631.19/231.69
POL(and(x1, x2)) = x2    631.19/231.69
POL(c(x1)) = x1    631.19/231.69
POL(c(x1, x2)) = x1 + x2    631.19/231.69
POL(c6(x1)) = x1    631.19/231.69
POL(even(x1)) = 0    631.19/231.69
POL(false) = 0    631.19/231.69
POL(gr(x1, x2)) = x1    631.19/231.69
POL(p(x1)) = 0    631.19/231.69
POL(s(x1)) = [1]    631.19/231.69
POL(true) = [1]   
631.19/231.69
631.19/231.69

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.19/231.69
and(z0, false) → false 631.19/231.69
and(false, z0) → false 631.19/231.69
and(true, true) → true 631.19/231.69
even(0) → true 631.19/231.69
even(s(0)) → false 631.19/231.69
even(s(s(z0))) → even(z0) 631.19/231.69
gr(0, z0) → false 631.19/231.69
gr(s(z0), 0) → true 631.19/231.69
gr(s(z0), s(y)) → gr(z0, y) 631.19/231.69
p(0) → 0 631.19/231.69
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.19/231.69
631.19/231.69

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0)))
We considered the (Usable) Rules:

gr(s(z0), 0) → true 631.19/231.69
gr(0, z0) → false 631.19/231.69
and(z0, false) → false 631.19/231.69
and(false, z0) → false 631.19/231.69
and(true, true) → true 631.19/231.69
p(s(z0)) → z0 631.19/231.69
p(0) → 0 631.19/231.69
even(0) → true 631.19/231.69
even(s(0)) → false 631.19/231.69
even(s(s(z0))) → even(z0)
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.19/231.69

POL(0) = 0    631.19/231.69
POL(COND(x1, x2)) = [2]x2    631.19/231.69
POL(EVEN(x1)) = 0    631.19/231.69
POL(and(x1, x2)) = 0    631.19/231.69
POL(c(x1)) = x1    631.19/231.69
POL(c(x1, x2)) = x1 + x2    631.19/231.69
POL(c6(x1)) = x1    631.19/231.69
POL(even(x1)) = 0    631.19/231.69
POL(false) = 0    631.19/231.69
POL(gr(x1, x2)) = 0    631.19/231.69
POL(p(x1)) = x1    631.19/231.69
POL(s(x1)) = [4] + x1    631.19/231.69
POL(true) = 0   
631.19/231.69
631.19/231.69

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.19/231.69
and(z0, false) → false 631.19/231.69
and(false, z0) → false 631.19/231.69
and(true, true) → true 631.19/231.69
even(0) → true 631.19/231.69
even(s(0)) → false 631.19/231.69
even(s(s(z0))) → even(z0) 631.19/231.69
gr(0, z0) → false 631.19/231.69
gr(s(z0), 0) → true 631.19/231.69
gr(s(z0), s(y)) → gr(z0, y) 631.19/231.69
p(0) → 0 631.19/231.69
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.19/231.69
631.19/231.69

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) by

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0), EVEN(s(0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0))))
631.19/231.69
631.19/231.69

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.19/231.69
and(z0, false) → false 631.19/231.69
and(false, z0) → false 631.19/231.69
and(true, true) → true 631.19/231.69
even(0) → true 631.19/231.69
even(s(0)) → false 631.19/231.69
even(s(s(z0))) → even(z0) 631.19/231.69
gr(0, z0) → false 631.19/231.69
gr(s(z0), 0) → true 631.19/231.69
gr(s(z0), s(y)) → gr(z0, y) 631.19/231.69
p(0) → 0 631.19/231.69
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.19/231.69
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0), EVEN(s(0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.19/231.69
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.19/231.69
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.19/231.69
631.19/231.69

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.19/231.69
631.19/231.69

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.19/231.69
and(z0, false) → false 631.19/231.69
and(false, z0) → false 631.19/231.69
and(true, true) → true 631.19/231.69
even(0) → true 631.19/231.69
even(s(0)) → false 631.19/231.69
even(s(s(z0))) → even(z0) 631.19/231.69
gr(0, z0) → false 631.19/231.69
gr(s(z0), 0) → true 631.19/231.69
gr(s(z0), s(y)) → gr(z0, y) 631.19/231.69
p(0) → 0 631.19/231.69
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.19/231.69
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.19/231.69
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, s(z0)) → c(COND(and(even(s(z0)), true), p(s(z0))), EVEN(s(z0))) by

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))), EVEN(s(0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0))))
631.58/231.74
631.58/231.74

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))), EVEN(s(0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))), EVEN(s(0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(23) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.74
631.58/231.74

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(25) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0)))
We considered the (Usable) Rules:

and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
and(z0, false) → false 631.58/231.74
p(s(z0)) → z0 631.58/231.74
p(0) → 0 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(0, z0) → false
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.58/231.74

POL(0) = 0    631.58/231.74
POL(COND(x1, x2)) = [4]x2    631.58/231.74
POL(EVEN(x1)) = 0    631.58/231.74
POL(and(x1, x2)) = 0    631.58/231.74
POL(c(x1)) = x1    631.58/231.74
POL(c(x1, x2)) = x1 + x2    631.58/231.74
POL(c6(x1)) = x1    631.58/231.74
POL(even(x1)) = 0    631.58/231.74
POL(false) = 0    631.58/231.74
POL(gr(x1, x2)) = [2]    631.58/231.74
POL(p(x1)) = x1    631.58/231.74
POL(s(x1)) = [4] + x1    631.58/231.74
POL(true) = 0   
631.58/231.74
631.58/231.74

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
We considered the (Usable) Rules:

and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
and(z0, false) → false 631.58/231.74
p(s(z0)) → z0 631.58/231.74
p(0) → 0 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(0, z0) → false
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.58/231.74

POL(0) = 0    631.58/231.74
POL(COND(x1, x2)) = x1    631.58/231.74
POL(EVEN(x1)) = 0    631.58/231.74
POL(and(x1, x2)) = x1    631.58/231.74
POL(c(x1)) = x1    631.58/231.74
POL(c(x1, x2)) = x1 + x2    631.58/231.74
POL(c6(x1)) = x1    631.58/231.74
POL(even(x1)) = [2]    631.58/231.74
POL(false) = 0    631.58/231.74
POL(gr(x1, x2)) = [4]x1    631.58/231.74
POL(p(x1)) = [4]    631.58/231.74
POL(s(x1)) = [2]    631.58/231.74
POL(true) = [2]   
631.58/231.74
631.58/231.74

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), p(s(s(z0)))), EVEN(s(s(z0)))) by

COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
631.58/231.74
631.58/231.74

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(31) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
We considered the (Usable) Rules:

even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(0, z0) → false 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
p(s(z0)) → z0 631.58/231.74
p(0) → 0
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.58/231.74

POL(0) = [2]    631.58/231.74
POL(COND(x1, x2)) = [2]    631.58/231.74
POL(EVEN(x1)) = 0    631.58/231.74
POL(and(x1, x2)) = 0    631.58/231.74
POL(c(x1)) = x1    631.58/231.74
POL(c(x1, x2)) = x1 + x2    631.58/231.74
POL(c6(x1)) = x1    631.58/231.74
POL(even(x1)) = [2] + [2]x1    631.58/231.74
POL(false) = [4]    631.58/231.74
POL(gr(x1, x2)) = [2] + [4]x1    631.58/231.74
POL(p(x1)) = 0    631.58/231.74
POL(s(x1)) = 0    631.58/231.74
POL(true) = 0   
631.58/231.74
631.58/231.74

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(33) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0))))
We considered the (Usable) Rules:

even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(0, z0) → false 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
p(s(z0)) → z0 631.58/231.74
p(0) → 0
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.58/231.74

POL(0) = 0    631.58/231.74
POL(COND(x1, x2)) = x2    631.58/231.74
POL(EVEN(x1)) = 0    631.58/231.74
POL(and(x1, x2)) = 0    631.58/231.74
POL(c(x1)) = x1    631.58/231.74
POL(c(x1, x2)) = x1 + x2    631.58/231.74
POL(c6(x1)) = x1    631.58/231.74
POL(even(x1)) = 0    631.58/231.74
POL(false) = 0    631.58/231.74
POL(gr(x1, x2)) = [2]x1    631.58/231.74
POL(p(x1)) = x1    631.58/231.74
POL(s(x1)) = [1] + x1    631.58/231.74
POL(true) = 0   
631.58/231.74
631.58/231.74

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(35) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
We considered the (Usable) Rules:

even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(0, z0) → false 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
p(s(z0)) → z0 631.58/231.74
p(0) → 0
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.58/231.74

POL(0) = 0    631.58/231.74
POL(COND(x1, x2)) = [2] + [4]x1    631.58/231.74
POL(EVEN(x1)) = 0    631.58/231.74
POL(and(x1, x2)) = x1    631.58/231.74
POL(c(x1)) = x1    631.58/231.74
POL(c(x1, x2)) = x1 + x2    631.58/231.74
POL(c6(x1)) = x1    631.58/231.74
POL(even(x1)) = [4]    631.58/231.74
POL(false) = 0    631.58/231.74
POL(gr(x1, x2)) = 0    631.58/231.74
POL(p(x1)) = 0    631.58/231.74
POL(s(x1)) = 0    631.58/231.74
POL(true) = [4]   
631.58/231.74
631.58/231.74

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) by

COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0))
631.58/231.74
631.58/231.74

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, 0) → c(COND(and(even(0), false), p(0))) by

COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(false, p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0)))
631.58/231.74
631.58/231.74

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(false, p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0)))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c

631.58/231.74
631.58/231.74

(41) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.74
631.58/231.74

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.74
COND(true, 0) → c
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), p(0))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), gr(s(z0), 0)), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.74
631.58/231.74

(43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

COND(true, 0) → c
631.58/231.74
631.58/231.74

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.74
COND(true, 0) → c
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.74
631.58/231.74

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) by

COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0)))
631.58/231.74
631.58/231.74

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.74
COND(true, 0) → c
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), p(0))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.74
631.58/231.74

(47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

COND(true, 0) → c
631.58/231.74
631.58/231.74

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.74
COND(true, 0) → c
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.74
631.58/231.74

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) by

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(0)) → c(COND(false, p(s(0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0))))
631.58/231.74
631.58/231.74

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.74
COND(true, 0) → c 631.58/231.74
COND(true, s(0)) → c(COND(false, p(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.74
631.58/231.74

(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.74
631.58/231.74

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.74
COND(true, 0) → c 631.58/231.74
COND(true, s(0)) → c
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), p(s(0)))) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.74
631.58/231.74

(53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.74
COND(true, 0) → c
631.58/231.74
631.58/231.74

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.74
COND(true, 0) → c 631.58/231.74
COND(true, s(0)) → c
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.74
631.58/231.74

(55) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) by COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
631.58/231.74
631.58/231.74

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.74
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.74
COND(true, 0) → c 631.58/231.74
COND(true, s(0)) → c 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.74
631.58/231.74

(57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.74
COND(true, 0) → c
631.58/231.74
631.58/231.74

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.74
and(z0, false) → false 631.58/231.74
and(false, z0) → false 631.58/231.74
and(true, true) → true 631.58/231.74
even(0) → true 631.58/231.74
even(s(0)) → false 631.58/231.74
even(s(s(z0))) → even(z0) 631.58/231.74
gr(0, z0) → false 631.58/231.74
gr(s(z0), 0) → true 631.58/231.74
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.74
p(0) → 0 631.58/231.74
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.74
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.74
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.74
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.74
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.74
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.74
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.74
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.76
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.76
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.76
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.76
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.76
COND(true, 0) → c 631.58/231.76
COND(true, s(0)) → c 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.76
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.76
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.76
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.76
631.58/231.76

(59) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) by COND(true, s(0)) → c(COND(and(false, true), 0))
631.58/231.76
631.58/231.76

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.76
and(z0, false) → false 631.58/231.76
and(false, z0) → false 631.58/231.76
and(true, true) → true 631.58/231.76
even(0) → true 631.58/231.76
even(s(0)) → false 631.58/231.76
even(s(s(z0))) → even(z0) 631.58/231.76
gr(0, z0) → false 631.58/231.76
gr(s(z0), 0) → true 631.58/231.76
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.76
p(0) → 0 631.58/231.76
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.76
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.76
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.76
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.76
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.76
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.76
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.76
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.76
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.76
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.76
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.76
COND(true, 0) → c 631.58/231.76
COND(true, s(0)) → c 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.76
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.76
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.76
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.76
631.58/231.76

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.76
COND(true, 0) → c
631.58/231.76
631.58/231.76

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.76
and(z0, false) → false 631.58/231.76
and(false, z0) → false 631.58/231.76
and(true, true) → true 631.58/231.76
even(0) → true 631.58/231.76
even(s(0)) → false 631.58/231.76
even(s(s(z0))) → even(z0) 631.58/231.76
gr(0, z0) → false 631.58/231.76
gr(s(z0), 0) → true 631.58/231.76
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.76
p(0) → 0 631.58/231.76
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.76
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.76
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.76
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.76
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.76
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.76
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.76
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.76
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.76
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.76
COND(true, 0) → c 631.58/231.76
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.76
COND(true, s(0)) → c 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.76
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.76
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.76
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.76
631.58/231.76

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, s(0)) → c(COND(and(false, true), 0)) by

COND(true, s(0)) → c(COND(false, 0))
631.58/231.76
631.58/231.76

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.76
and(z0, false) → false 631.58/231.76
and(false, z0) → false 631.58/231.76
and(true, true) → true 631.58/231.76
even(0) → true 631.58/231.76
even(s(0)) → false 631.58/231.76
even(s(s(z0))) → even(z0) 631.58/231.76
gr(0, z0) → false 631.58/231.76
gr(s(z0), 0) → true 631.58/231.76
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.76
p(0) → 0 631.58/231.76
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.76
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.76
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.76
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.76
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.77
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.77
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.77
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.77
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.77
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.77
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.77
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.77
COND(true, 0) → c 631.58/231.77
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.77
COND(true, s(0)) → c 631.58/231.77
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.77
COND(true, s(0)) → c(COND(false, 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.77
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.77
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.77
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.77
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.77
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.77
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.77
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.77
631.58/231.77

(65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.77
631.58/231.77

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.77
and(z0, false) → false 631.58/231.77
and(false, z0) → false 631.58/231.77
and(true, true) → true 631.58/231.77
even(0) → true 631.58/231.77
even(s(0)) → false 631.58/231.77
even(s(s(z0))) → even(z0) 631.58/231.77
gr(0, z0) → false 631.58/231.77
gr(s(z0), 0) → true 631.58/231.77
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.77
p(0) → 0 631.58/231.77
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.77
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.77
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.77
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.77
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.78
COND(true, 0) → c
631.58/231.78
631.58/231.78

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(69) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) by COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
631.58/231.78
631.58/231.78

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), p(s(s(z0)))), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.78
COND(true, 0) → c
631.58/231.78
631.58/231.78

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0)))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(73) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
We considered the (Usable) Rules:

even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
and(z0, false) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(0, z0) → false 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.58/231.78

POL(0) = 0    631.58/231.78
POL(COND(x1, x2)) = [2]x2    631.58/231.78
POL(EVEN(x1)) = 0    631.58/231.78
POL(and(x1, x2)) = 0    631.58/231.78
POL(c) = 0    631.58/231.78
POL(c(x1)) = x1    631.58/231.78
POL(c(x1, x2)) = x1 + x2    631.58/231.78
POL(c6(x1)) = x1    631.58/231.78
POL(even(x1)) = 0    631.58/231.78
POL(false) = 0    631.58/231.78
POL(gr(x1, x2)) = 0    631.58/231.78
POL(p(x1)) = x1    631.58/231.78
POL(s(x1)) = [1] + x1    631.58/231.78
POL(true) = 0   
631.58/231.78
631.58/231.78

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(75) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) by COND(true, s(0)) → c(COND(and(false, true), 0))
631.58/231.78
631.58/231.78

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.78
COND(true, 0) → c
631.58/231.78
631.58/231.78

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, s(0)) → c(COND(and(false, true), 0)) by

COND(true, s(0)) → c(COND(false, 0))
631.58/231.78
631.58/231.78

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(0)) → c(COND(false, 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.78
631.58/231.78

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.78
COND(true, 0) → c
631.58/231.78
631.58/231.78

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(85) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(x0))) → c(COND(and(even(x0), gr(s(s(x0)), 0)), s(x0)), EVEN(s(s(x0)))) by COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
631.58/231.78
631.58/231.78

(86) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), gr(s(s(z0)), 0)), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.78
COND(true, 0) → c
631.58/231.78
631.58/231.78

(88) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(89) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(x0))) → c(COND(and(even(x0), true), p(s(s(x0)))), EVEN(s(s(x0)))) by COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
631.58/231.78
631.58/231.78

(90) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.78
COND(true, 0) → c
631.58/231.78
631.58/231.78

(92) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(93) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), p(s(s(0)))), EVEN(s(s(0)))) by COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
631.58/231.78
631.58/231.78

(94) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.78
COND(true, 0) → c
631.58/231.78
631.58/231.78

(96) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(97) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
We considered the (Usable) Rules:

gr(s(z0), 0) → true 631.58/231.78
gr(0, z0) → false 631.58/231.78
and(z0, false) → false 631.58/231.78
and(true, true) → true 631.58/231.78
and(false, z0) → false 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
p(s(z0)) → z0 631.58/231.78
p(0) → 0
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.58/231.78

POL(0) = 0    631.58/231.78
POL(COND(x1, x2)) = [4] + x2    631.58/231.78
POL(EVEN(x1)) = 0    631.58/231.78
POL(and(x1, x2)) = 0    631.58/231.78
POL(c) = 0    631.58/231.78
POL(c(x1)) = x1    631.58/231.78
POL(c(x1, x2)) = x1 + x2    631.58/231.78
POL(c6(x1)) = x1    631.58/231.78
POL(even(x1)) = 0    631.58/231.78
POL(false) = 0    631.58/231.78
POL(gr(x1, x2)) = 0    631.58/231.78
POL(p(x1)) = x1    631.58/231.78
POL(s(x1)) = [2] + x1    631.58/231.78
POL(true) = 0   
631.58/231.78
631.58/231.78

(98) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(99) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) by COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0)))))
631.58/231.78
631.58/231.78

(100) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0)))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), p(s(s(s(0))))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.78
COND(true, 0) → c
631.58/231.78
631.58/231.78

(102) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0)))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(103) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), p(s(s(s(s(z0)))))), EVEN(s(s(s(s(z0)))))) by COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
631.58/231.78
631.58/231.78

(104) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.78
COND(true, 0) → c
631.58/231.78
631.58/231.78

(106) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(107) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
We considered the (Usable) Rules:

even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(0, z0) → false 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
p(s(z0)) → z0 631.58/231.78
p(0) → 0
And the Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.58/231.78

POL(0) = 0    631.58/231.78
POL(COND(x1, x2)) = [4]x2    631.58/231.78
POL(EVEN(x1)) = 0    631.58/231.78
POL(and(x1, x2)) = 0    631.58/231.78
POL(c) = 0    631.58/231.78
POL(c(x1)) = x1    631.58/231.78
POL(c(x1, x2)) = x1 + x2    631.58/231.78
POL(c6(x1)) = x1    631.58/231.78
POL(even(x1)) = 0    631.58/231.78
POL(false) = 0    631.58/231.78
POL(gr(x1, x2)) = 0    631.58/231.78
POL(p(x1)) = x1    631.58/231.78
POL(s(x1)) = [4] + x1    631.58/231.78
POL(true) = 0   
631.58/231.78
631.58/231.78

(108) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0))))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.78
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.78
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.78
631.58/231.78

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, 0) → c(COND(and(even(0), false), 0)) by COND(true, 0) → c(COND(and(true, false), 0))
631.58/231.78
631.58/231.78

(110) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.78
and(z0, false) → false 631.58/231.78
and(false, z0) → false 631.58/231.78
and(true, true) → true 631.58/231.78
even(0) → true 631.58/231.78
even(s(0)) → false 631.58/231.78
even(s(s(z0))) → even(z0) 631.58/231.78
gr(0, z0) → false 631.58/231.78
gr(s(z0), 0) → true 631.58/231.78
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.78
p(0) → 0 631.58/231.78
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.78
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.78
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.78
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.78
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.78
COND(true, 0) → c 631.58/231.78
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.78
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.78
COND(true, s(0)) → c 631.58/231.78
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0))))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.79
631.58/231.79

(111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(112) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.79
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0))))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.79
631.58/231.79

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, 0) → c(COND(and(true, false), 0)) by

COND(true, 0) → c(COND(false, 0))
631.58/231.79
631.58/231.79

(114) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.79
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0))))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, 0) → c(COND(false, 0))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.79
631.58/231.79

(115) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.79
631.58/231.79

(116) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.79
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0))))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.79
631.58/231.79

(117) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(118) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

EVEN(s(s(z0))) → c6(EVEN(z0)) 631.58/231.79
COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0))))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
S tuples:

EVEN(s(s(z0))) → c6(EVEN(z0))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

EVEN, COND

Compound Symbols:

c6, c, c, c

631.58/231.79
631.58/231.79

(119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace EVEN(s(s(z0))) → c6(EVEN(z0)) by

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
631.58/231.79
631.58/231.79

(120) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))), EVEN(s(s(s(0))))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)), EVEN(s(s(0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts
631.58/231.79
631.58/231.79

(122) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(123) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(124) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) by COND(true, 0) → c(COND(and(true, false), 0))
631.58/231.79
631.58/231.79

(126) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(127) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(128) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(129) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, 0) → c(COND(and(true, false), 0)) by

COND(true, 0) → c(COND(false, 0))
631.58/231.79
631.58/231.79

(130) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(false, 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.79
631.58/231.79

(132) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(134) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(even(0), false), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(135) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, 0) → c(COND(and(even(0), false), 0)) by COND(true, 0) → c(COND(and(true, false), 0))
631.58/231.79
631.58/231.79

(136) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(138) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(139) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, 0) → c(COND(and(true, false), 0)) by

COND(true, 0) → c(COND(false, 0))
631.58/231.79
631.58/231.79

(140) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(false, 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.79
631.58/231.79

(142) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(144) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(145) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, 0) → c(COND(and(true, false), p(0))) by COND(true, 0) → c(COND(and(true, false), 0))
631.58/231.79
631.58/231.79

(146) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(148) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), p(0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(149) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, 0) → c(COND(and(true, false), p(0))) by

COND(true, 0) → c(COND(and(true, false), 0)) 631.58/231.79
COND(true, 0) → c(COND(false, p(0)))
631.58/231.79
631.58/231.79

(150) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0)) 631.58/231.79
COND(true, 0) → c(COND(false, p(0)))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(151) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.79
631.58/231.79

(152) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(153) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(154) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(155) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) by

COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(false, p(s(0))))
631.58/231.79
631.58/231.79

(156) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(false, p(s(0))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.79
631.58/231.79

(158) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), p(s(0)))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(160) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(161) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, 0) → c(COND(and(true, false), 0)) by

COND(true, 0) → c(COND(false, 0))
631.58/231.79
631.58/231.79

(162) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, 0) → c(COND(false, 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.79
631.58/231.79

(164) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(165) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(166) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(167) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, 0) → c(COND(and(true, false), 0)) by

COND(true, 0) → c(COND(false, 0))
631.58/231.79
631.58/231.79

(168) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, 0) → c(COND(false, 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(169) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.79
631.58/231.79

(170) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(171) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(172) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(173) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, 0) → c(COND(and(true, gr(0, 0)), 0)) by COND(true, 0) → c(COND(and(true, false), 0))
631.58/231.79
631.58/231.79

(174) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(175) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c
631.58/231.79
631.58/231.79

(176) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, 0) → c(COND(and(true, false), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(177) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace COND(true, 0) → c(COND(and(true, false), 0)) by

COND(true, 0) → c(COND(false, 0))
631.58/231.79
631.58/231.79

(178) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, 0) → c(COND(false, 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(179) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
631.58/231.79
631.58/231.79

(180) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(181) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0))
631.58/231.79
631.58/231.79

(182) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(183) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(0)) → c(COND(and(false, gr(s(0), 0)), 0)) by COND(true, s(0)) → c(COND(and(false, true), 0))
631.58/231.79
631.58/231.79

(184) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(185) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
631.58/231.79
631.58/231.79

(186) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(187) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) by COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), true), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
631.58/231.79
631.58/231.79

(188) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), true), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), gr(s(s(s(s(z0)))), 0)), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(189) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
631.58/231.79
631.58/231.79

(190) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), true), s(s(s(z0)))), EVEN(s(s(s(s(z0))))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(191) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0))) by COND(true, s(s(0))) → c(COND(and(true, true), s(0)))
631.58/231.79
631.58/231.79

(192) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), true), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, true), s(0)))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, gr(s(s(0)), 0)), s(0)))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(193) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
631.58/231.79
631.58/231.79

(194) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), true), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, true), s(0)))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(195) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(s(0)))) → c(COND(and(false, gr(s(s(s(0))), 0)), s(s(0)))) by COND(true, s(s(s(0)))) → c(COND(and(false, true), s(s(0))))
631.58/231.79
631.58/231.79

(196) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), true), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, true), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, true), s(s(0))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(197) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

COND(true, s(0)) → c 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0))
631.58/231.79
631.58/231.79

(198) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), true), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, true), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, true), s(s(0))))
S tuples:

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(199) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
We considered the (Usable) Rules:

and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0)
And the Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), true), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, true), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, true), s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 631.58/231.79

POL(0) = [3]    631.58/231.79
POL(COND(x1, x2)) = [3]x2 + x22    631.58/231.79
POL(EVEN(x1)) = x1    631.58/231.79
POL(and(x1, x2)) = 0    631.58/231.79
POL(c) = 0    631.58/231.79
POL(c(x1)) = x1    631.58/231.79
POL(c(x1, x2)) = x1 + x2    631.58/231.79
POL(c6(x1)) = x1    631.58/231.79
POL(even(x1)) = 0    631.58/231.79
POL(false) = 0    631.58/231.79
POL(s(x1)) = [2] + x1    631.58/231.79
POL(true) = 0   
631.58/231.79
631.58/231.79

(200) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(and(even(z0), gr(z0, 0)), p(z0)) 631.58/231.79
and(z0, false) → false 631.58/231.79
and(false, z0) → false 631.58/231.79
and(true, true) → true 631.58/231.79
even(0) → true 631.58/231.79
even(s(0)) → false 631.58/231.79
even(s(s(z0))) → even(z0) 631.58/231.79
gr(0, z0) → false 631.58/231.79
gr(s(z0), 0) → true 631.58/231.79
gr(s(z0), s(y)) → gr(z0, y) 631.58/231.79
p(0) → 0 631.58/231.79
p(s(z0)) → z0
Tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, 0) → c 631.58/231.79
COND(true, s(0)) → c 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0)))) 631.58/231.79
COND(true, s(0)) → c(COND(and(false, true), 0)) 631.58/231.79
COND(true, s(s(s(s(z0))))) → c(COND(and(even(z0), true), s(s(s(z0)))), EVEN(s(s(s(s(z0)))))) 631.58/231.79
COND(true, s(s(0))) → c(COND(and(true, true), s(0))) 631.58/231.79
COND(true, s(s(s(0)))) → c(COND(and(false, true), s(s(0))))
S tuples:none
K tuples:

COND(true, s(z0)) → c(COND(and(even(s(z0)), true), z0), EVEN(s(z0))) 631.58/231.79
COND(true, s(s(x0))) → c(EVEN(s(s(x0)))) 631.58/231.79
COND(true, s(s(z0))) → c(COND(and(even(z0), true), s(z0)), EVEN(s(s(z0)))) 631.58/231.79
EVEN(s(s(s(s(y0))))) → c6(EVEN(s(s(y0))))
Defined Rule Symbols:

cond, and, even, gr, p

Defined Pair Symbols:

COND, EVEN

Compound Symbols:

c, c, c, c6

631.58/231.79
631.58/231.79

(201) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
631.58/231.79
631.58/231.79

(202) BOUNDS(O(1), O(1))

631.58/231.79
631.58/231.79
631.88/231.84 EOF