Runtime Complexity (innermost) proof of /tmp/tmpP4RK

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtUsableRulesProof (⇔, 0 ms)

↳8 CdtProblem

↳9 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 204 ms)

↳10 CdtProblem

↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CdtProblem

↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳14 CdtProblem

↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 77 ms)

↳16 CdtProblem

↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 38 ms)

↳18 CdtProblem

↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳20 CdtProblem

↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳22 CdtProblem

↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 417 ms)

↳24 CdtProblem

↳25 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 935 ms)

↳26 CdtProblem

↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳28 CdtProblem

↳29 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 117 ms)

↳30 CdtProblem

↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳32 CdtProblem

↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳34 CdtProblem

↳35 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 85 ms)

↳36 CdtProblem

↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳38 CdtProblem

↳39 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳40 CdtProblem

↳41 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 223 ms)

↳42 CdtProblem

↳43 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳44 CdtProblem

↳45 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 37 ms)

↳46 CdtProblem

↳47 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 370 ms)

↳48 CdtProblem

↳49 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳50 CdtProblem

↳51 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳52 CdtProblem

↳53 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 857 ms)

↳54 CdtProblem

↳55 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳56 CdtProblem

↳57 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳58 CdtProblem

↳59 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 150 ms)

↳60 CdtProblem

↳61 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳62 CdtProblem

↳63 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳64 CdtProblem

↳65 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳66 CdtProblem

↳67 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳68 CdtProblem

↳69 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳70 CdtProblem

↳71 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳72 CdtProblem

↳73 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳74 CdtProblem

↳75 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳76 CdtProblem

↳77 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳78 CdtProblem

↳79 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳80 CdtProblem

↳81 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳82 CdtProblem

↳83 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳84 CdtProblem

↳85 CdtUsableRulesProof (⇔, 0 ms)

↳86 CdtProblem

↳87 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳88 CdtProblem

↳89 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 513 ms)

↳90 CdtProblem

↳91 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳92 CdtProblem

↳93 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳94 CdtProblem

↳95 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳96 CdtProblem

↳97 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳98 CdtProblem

↳99 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳100 CdtProblem

↳101 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 128 ms)

↳102 CdtProblem

↳103 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳104 CdtProblem

↳105 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳106 CdtProblem

↳107 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳108 CdtProblem

↳109 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 13 ms)

↳110 CdtProblem

↳111 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳112 CdtProblem

↳113 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳114 CdtProblem

↳115 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳116 CdtProblem

↳117 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳118 CdtProblem

↳119 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳120 CdtProblem

↳121 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳122 CdtProblem

↳123 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 8 ms)

↳124 CdtProblem

↳125 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

↳126 BOUNDS(1, 1)

(0) Obligation:

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

cond1(true, x, y) → cond2(gr(y, 0), x, y)
cond2(true, x, y) → cond2(gr(y, 0), p(x), p(y))
cond2(false, x, y) → cond1(and(eq(x, y), gr(x, 0)), x, y)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
eq(0, 0) → true
eq(s(x), 0) → false
eq(0, s(x)) → false
eq(s(x), s(y)) → eq(x, y)
and(true, true) → true
and(false, x) → false
and(x, false) → false

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1) → cond2(gr(z1, 0), z0, z1)
cond2(true, z0, z1) → cond2(gr(z1, 0), p(z0), p(z1))
cond2(false, z0, z1) → cond1(and(eq(z0, z1), gr(z0, 0)), z0, z1)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)
and(true, true) → true
and(false, z0) → false
and(z0, false) → false

Tuples:

COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1), GR(z1, 0))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)), GR(z1, 0), P(z0), P(z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), AND(eq(z0, z1), gr(z0, 0)), EQ(z0, z1), GR(z0, 0))
GR(0, z0) → c3
GR(s(z0), 0) → c4
GR(s(z0), s(z1)) → c5(GR(z0, z1))
P(0) → c6
P(s(z0)) → c7
EQ(0, 0) → c8
EQ(s(z0), 0) → c9
EQ(0, s(z0)) → c10
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
AND(true, true) → c12
AND(false, z0) → c13
AND(z0, false) → c14

S tuples:

COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1), GR(z1, 0))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)), GR(z1, 0), P(z0), P(z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), AND(eq(z0, z1), gr(z0, 0)), EQ(z0, z1), GR(z0, 0))
GR(0, z0) → c3
GR(s(z0), 0) → c4
GR(s(z0), s(z1)) → c5(GR(z0, z1))
P(0) → c6
P(s(z0)) → c7
EQ(0, 0) → c8
EQ(s(z0), 0) → c9
EQ(0, s(z0)) → c10
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
AND(true, true) → c12
AND(false, z0) → c13
AND(z0, false) → c14

K tuples:none
Defined Rule Symbols:

cond1, cond2, gr, p, eq, and

Defined Pair Symbols:

COND1, COND2, GR, P, EQ, AND

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14

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

Removed 10 trailing nodes:

AND(true, true) → c12
AND(false, z0) → c13
GR(0, z0) → c3
EQ(s(z0), 0) → c9
EQ(0, 0) → c8
AND(z0, false) → c14
P(0) → c6
P(s(z0)) → c7
EQ(0, s(z0)) → c10
GR(s(z0), 0) → c4

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1) → cond2(gr(z1, 0), z0, z1)
cond2(true, z0, z1) → cond2(gr(z1, 0), p(z0), p(z1))
cond2(false, z0, z1) → cond1(and(eq(z0, z1), gr(z0, 0)), z0, z1)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)
and(true, true) → true
and(false, z0) → false
and(z0, false) → false

Tuples:

COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1), GR(z1, 0))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)), GR(z1, 0), P(z0), P(z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), AND(eq(z0, z1), gr(z0, 0)), EQ(z0, z1), GR(z0, 0))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))

S tuples:

COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1), GR(z1, 0))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)), GR(z1, 0), P(z0), P(z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), AND(eq(z0, z1), gr(z0, 0)), EQ(z0, z1), GR(z0, 0))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))

K tuples:none
Defined Rule Symbols:

cond1, cond2, gr, p, eq, and

Defined Pair Symbols:

COND1, COND2, GR, EQ

Compound Symbols:

c, c1, c2, c5, c11

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

Removed 6 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1) → cond2(gr(z1, 0), z0, z1)
cond2(true, z0, z1) → cond2(gr(z1, 0), p(z0), p(z1))
cond2(false, z0, z1) → cond1(and(eq(z0, z1), gr(z0, 0)), z0, z1)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)
and(true, true) → true
and(false, z0) → false
and(z0, false) → false

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))

K tuples:none
Defined Rule Symbols:

cond1, cond2, gr, p, eq, and

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

cond1(true, z0, z1) → cond2(gr(z1, 0), z0, z1)
cond2(true, z0, z1) → cond2(gr(z1, 0), p(z0), p(z1))
cond2(false, z0, z1) → cond1(and(eq(z0, z1), gr(z0, 0)), z0, z1)
gr(s(z0), s(z1)) → gr(z0, z1)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))

K tuples:none
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = 0
POL(COND2(x₁, x₂, x₃)) = 0
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = [4]x₁ + [4]x₂
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [2] + [4]x₁
POL(false) = [2]
POL(gr(x₁, x₂)) = [5]x₁ + [2]x₂
POL(p(x₁)) = [3]
POL(s(x₁)) = [4] + x₁
POL(true) = 0

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2

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

Use narrowing to replace COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1)) by

COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND2, COND1

Compound Symbols:

c5, c11, c1, c2, c

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

Use narrowing to replace COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1))) by

COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND2, COND1

Compound Symbols:

c5, c11, c2, c, c1

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [4]x₂
POL(COND2(x₁, x₂, x₃)) = [4]x₂
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = [3]x₁ + [4]x₂
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = 0
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(p(x₁)) = x₁
POL(s(x₁)) = [2] + x₁
POL(true) = [1]

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND2, COND1

Compound Symbols:

c5, c11, c2, c, c1

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [2]x₃
POL(COND2(x₁, x₂, x₃)) = [2]x₃
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = [5]x₁ + [4]x₂
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [2]x₂
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(p(x₁)) = x₁
POL(s(x₁)) = [4] + x₁
POL(true) = 0

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND2, COND1

Compound Symbols:

c5, c11, c2, c, c1

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

Use narrowing to replace COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1)) by

COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1), EQ(0, x1))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0), EQ(0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0), EQ(s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)), EQ(0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1), EQ(0, x1))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0), EQ(0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0), EQ(s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)), EQ(0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1), EQ(0, x1))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0), EQ(0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0), EQ(s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)), EQ(0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2

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

Removed 4 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))

We considered the (Usable) Rules:

gr(0, z0) → false
and(true, true) → true
gr(s(z0), 0) → true
and(false, z0) → false
and(z0, false) → false

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [3] + [2]x₁
POL(COND2(x₁, x₂, x₃)) = [5]
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = 0
POL(false) = 0
POL(gr(x₁, x₂)) = x₁
POL(p(x₁)) = 0
POL(s(x₁)) = [1]
POL(true) = [1]

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))

We considered the (Usable) Rules:

eq(s(z0), 0) → false
eq(s(z0), s(z1)) → eq(z0, z1)
eq(0, s(z0)) → false
and(true, true) → true
eq(0, 0) → true
and(false, z0) → false
and(z0, false) → false

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₁
POL(COND2(x₁, x₂, x₃)) = [1]
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [1]
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(p(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = [1]

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0)) by

COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(29) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [2]x₂
POL(COND2(x₁, x₂, x₃)) = [2]x₂
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = [4]x₁ + x₂
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [4]x₁
POL(false) = 0
POL(gr(x₁, x₂)) = x₂
POL(p(x₁)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = [2]

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0)) by

COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1))) by

COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, 0, s(z0)) → c1(COND2(gr(s(z0), 0), 0, z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(z0)) → c1(COND2(gr(s(z0), 0), 0, z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₃
POL(COND2(x₁, x₂, x₃)) = x₃
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = [2]x₁ + [3]x₂
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [3] + [4]x₂
POL(false) = 0
POL(gr(x₁, x₂)) = [3]x₂
POL(p(x₁)) = x₁
POL(s(x₁)) = [4] + x₁
POL(true) = 0

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1))) by

COND2(true, s(x0), 0) → c1(COND2(gr(0, 0), x0, 0))
COND2(true, s(x0), s(z0)) → c1(COND2(gr(s(z0), 0), x0, z0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0))) by

COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, s(z0), 0) → c1(COND2(false, z0, p(0)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(z0), 0) → c1(COND2(false, z0, p(0)))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₂
POL(COND2(x₁, x₂, x₃)) = x₂
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = 0
POL(false) = 0
POL(gr(x₁, x₂)) = [2]x₂
POL(p(x₁)) = x₁
POL(s(x₁)) = [2] + x₁
POL(true) = 0

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0)))) by

COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, s(x1)) → c1(COND2(true, 0, p(s(x1))))
COND2(true, s(z0), s(x1)) → c1(COND2(true, z0, p(s(x1))))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, s(z0), s(x1)) → c1(COND2(true, z0, p(s(x1))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₂
POL(COND2(x₁, x₂, x₃)) = x₂
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = [5]x₂
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [2] + [3]x₂
POL(false) = 0
POL(gr(x₁, x₂)) = [1] + [2]x₂
POL(p(x₁)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

(47) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [2]x₃
POL(COND2(x₁, x₂, x₃)) = [2]x₃
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = [4]x₁ + [3]x₂
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = 0
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(p(x₁)) = x₁
POL(s(x₁)) = [2] + x₁
POL(true) = 0

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

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

Use narrowing to replace COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1)) by

COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0), EQ(s(z0), 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0), EQ(s(z0), 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0), EQ(s(z0), 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

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

Removed 1 trailing tuple parts

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

(53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

We considered the (Usable) Rules:

eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)
eq(0, 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₁ + x₂
POL(COND2(x₁, x₂, x₃)) = [1]
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [1]
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(p(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = [1]

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

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

The following tuples could be moved from S to K by knowledge propagation:

COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

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

Use narrowing to replace COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1))) by

COND2(false, s(z0), s(x1)) → c2(COND1(and(eq(z0, x1), true), s(z0), s(x1)), EQ(s(z0), s(x1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

(59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

We considered the (Usable) Rules:

eq(0, s(z0)) → false
eq(s(z0), 0) → false
eq(s(z0), s(z1)) → eq(z0, z1)
eq(0, 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [2] + [2]x₁
POL(COND2(x₁, x₂, x₃)) = [4]
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [1]
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(p(x₁)) = [2]
POL(s(x₁)) = 0
POL(true) = [1]

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Use narrowing to replace COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1)) by

COND2(false, 0, x0) → c2(COND1(false, 0, x0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, x0) → c2(COND1(false, 0, x0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Removed 1 trailing nodes:

COND2(false, 0, x0) → c2(COND1(false, 0, x0))

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Use narrowing to replace COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0)) by

COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Use narrowing to replace COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0)) by

COND2(false, s(x0), 0) → c2(COND1(false, s(x0), 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(false, s(x0), 0) → c2(COND1(false, s(x0), 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Removed 1 trailing nodes:

COND2(false, s(x0), 0) → c2(COND1(false, s(x0), 0))

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Use narrowing to replace COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0))) by

COND2(false, 0, s(x0)) → c2(COND1(false, 0, s(x0)))
COND2(false, 0, s(x0)) → c2(COND1(and(false, false), 0, s(x0)))

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(false, 0, s(x0)) → c2(COND1(false, 0, s(x0)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Removed 1 trailing nodes:

COND2(false, 0, s(x0)) → c2(COND1(false, 0, s(x0)))

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0)) by

COND2(true, 0, 0) → c1(COND2(false, 0, 0))

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0)) by

COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0)) by

COND2(true, 0, 0) → c1(COND2(false, 0, 0))

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, 0, 0) → c1(COND2(false, 0, p(0))) by

COND2(true, 0, 0) → c1(COND2(false, 0, 0))

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, s(x0), 0) → c1(COND2(gr(0, 0), x0, 0)) by

COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(85) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

gr(0, z0) → false

(86) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, 0) → c1(COND2(false, p(x0), 0)) by

COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(z0), 0) → c1(COND2(false, z0, 0))

(88) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(z0), 0) → c1(COND2(false, z0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₂
POL(COND2(x₁, x₂, x₃)) = x₂
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = [3]x₂
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [2]x₁ + [3]x₂
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(p(x₁)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(90) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1)) by

COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))

(92) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1)) by

COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))

(94) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

The following tuples could be moved from S to K by knowledge propagation:

COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

(96) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, 0, 0) → c1(COND2(false, 0, p(0))) by

COND2(true, 0, 0) → c1(COND2(false, 0, 0))

(98) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0)))) by

COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))

(100) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₃
POL(COND2(x₁, x₂, x₃)) = [2]x₁ + x₃
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = [3]x₁ + [4]x₂
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [2] + [4]x₁ + [2]x₂
POL(false) = 0
POL(gr(x₁, x₂)) = [2] + [2]x₁ + [5]x₂
POL(p(x₁)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(102) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0))) by

COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

(104) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Split RHS of tuples not part of any SCC

(106) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(z0), s(z1)) → c3(EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(0), s(0)) → c3(EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(0), s(s(z0))) → c3(EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), s(s(z1))) → c3(EQ(s(s(z0)), s(s(z1))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(z0), s(z1)) → c3(EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(0), s(0)) → c3(EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(0), s(s(z0))) → c3(EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), s(s(z1))) → c3(EQ(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c3

(107) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 5 leading nodes:

COND2(false, s(z0), s(z1)) → c3(EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(EQ(s(s(z0)), s(s(z1))))

(108) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))

S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c3

(109) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = 0
POL(COND2(x₁, x₂, x₃)) = 0
POL(EQ(x₁, x₂)) = x₂
POL(GR(x₁, x₂)) = [4]x₁ + [2]x₂
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [4] + x₁
POL(false) = 0
POL(gr(x₁, x₂)) = [3] + [2]x₁ + [4]x₂
POL(p(x₁)) = [3]
POL(s(x₁)) = [4] + x₁
POL(true) = 0

(110) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))

S tuples:

COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c3

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

Use narrowing to replace COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0)))) by

COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))

(112) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))

S tuples:

COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c3

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

Use narrowing to replace COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0)) by

COND2(false, s(x0), 0) → c2(COND1(false, s(x0), 0))

(114) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
COND2(false, s(x0), 0) → c2(COND1(false, s(x0), 0))

S tuples:

COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c3

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

Removed 1 trailing nodes:

COND2(false, s(x0), 0) → c2(COND1(false, s(x0), 0))

(116) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))

S tuples:

COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c3

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

Use narrowing to replace COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0)) by

COND2(false, 0, 0) → c2(COND1(false, 0, 0))

(118) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(false, 0, 0))

S tuples:

COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c3

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

Removed 3 trailing nodes:

COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(true, s(z0), 0) → c1(COND2(false, z0, 0))
COND2(false, 0, 0) → c2(COND1(false, 0, 0))

(120) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))

S tuples:

COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c3

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

Removed 3 trailing tuple parts

(122) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1

S tuples:

COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1

K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c3, c1

(123) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1

We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [2]x₃
POL(COND2(x₁, x₂, x₃)) = [1] + [3]x₁
POL(EQ(x₁, x₂)) = 0
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = [4] + [2]x₂
POL(c(x₁)) = x₁
POL(c1) = 0
POL(c1(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(eq(x₁, x₂)) = [3] + [2]x₁ + [3]x₂
POL(false) = [4]
POL(gr(x₁, x₂)) = [3] + [3]x₁ + [4]x₂
POL(p(x₁)) = [2] + [3]x₁
POL(s(x₁)) = [4]
POL(true) = [2]

(124) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
gr(s(z0), 0) → true
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
eq(0, 0) → true
eq(s(z0), 0) → false
eq(0, s(z0)) → false
eq(s(z0), s(z1)) → eq(z0, z1)

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1

S tuples:none
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, z0))
COND2(false, s(z0), s(z1)) → c3(COND1(and(eq(z0, z1), true), s(z0), s(z1)))
COND2(false, s(0), s(0)) → c3(COND1(and(true, gr(s(0), 0)), s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c3(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c3(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c3(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1

Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c3, c1

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

The set S is empty