Runtime Complexity (innermost) proof of /tmp/tmpiAIdgs/24.xml

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 12 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 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳10 CdtProblem

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

↳12 CdtProblem

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

↳14 CdtProblem

↳15 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳16 CdtProblem

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

↳18 CdtProblem

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

↳20 CdtProblem

↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 46 ms)

↳22 CdtProblem

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

↳24 CdtProblem

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

↳26 CdtProblem

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

↳28 CdtProblem

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

↳30 CdtProblem

↳31 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 31 ms)

↳32 CdtProblem

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

↳34 CdtProblem

↳35 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 33 ms)

↳36 CdtProblem

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

↳38 CdtProblem

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

↳40 CdtProblem

↳41 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 34 ms)

↳42 CdtProblem

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

↳44 CdtProblem

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

↳46 CdtProblem

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

↳48 CdtProblem

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

↳50 CdtProblem

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

↳52 CdtProblem

↳53 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 26 ms)

↳54 CdtProblem

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

↳56 CdtProblem

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

↳58 CdtProblem

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

↳60 CdtProblem

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

↳62 CdtProblem

↳63 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 24 ms)

↳64 CdtProblem

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

↳80 CdtProblem

↳81 CdtUsableRulesProof (⇔, 0 ms)

↳82 CdtProblem

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

↳84 CdtProblem

↳85 CdtUsableRulesProof (⇔, 0 ms)

↳86 CdtProblem

↳87 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)

↳88 CdtProblem

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

↳90 CdtProblem

↳91 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 3014 ms)

↳92 CdtProblem

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

↳94 CdtProblem

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

↳102 CdtProblem

↳103 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 1334 ms)

↳104 CdtProblem

↳105 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 1716 ms)

↳106 CdtProblem

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

↳108 CdtProblem

↳109 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 1729 ms)

↳110 CdtProblem

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

↳112 CdtProblem

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

↳114 CdtProblem

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

↳116 CdtProblem

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

↳118 BOUNDS(1, 1)

(0) Obligation:

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

The TRS R consists of the following rules:

cond1(true, x, y) → cond2(gr(x, y), x, y)
cond2(true, x, y) → cond3(gr(x, 0), x, y)
cond2(false, x, y) → cond4(gr(y, 0), x, y)
cond3(true, x, y) → cond3(gr(x, 0), p(x), y)
cond3(false, x, y) → cond1(and(gr(x, 0), gr(y, 0)), x, y)
cond4(true, x, y) → cond4(gr(y, 0), x, p(y))
cond4(false, x, y) → cond1(and(gr(x, 0), gr(y, 0)), x, y)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
and(true, true) → true
and(false, x) → false
and(x, false) → false
p(0) → 0
p(s(x)) → x

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(z0, z1), z0, z1)
cond2(true, z0, z1) → cond3(gr(z0, 0), z0, z1)
cond2(false, z0, z1) → cond4(gr(z1, 0), z0, z1)
cond3(true, z0, z1) → cond3(gr(z0, 0), p(z0), z1)
cond3(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
cond4(true, z0, z1) → cond4(gr(z1, 0), z0, p(z1))
cond4(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
p(0) → 0
p(s(z0)) → z0

Tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1), GR(z0, 0))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1), GR(z1, 0))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1), GR(z0, 0), P(z0))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)), GR(z1, 0), P(z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
GR(0, z0) → c7
GR(s(z0), 0) → c8
GR(s(z0), s(z1)) → c9(GR(z0, z1))
AND(true, true) → c10
AND(false, z0) → c11
AND(z0, false) → c12
P(0) → c13
P(s(z0)) → c14

S tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1), GR(z0, 0))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1), GR(z1, 0))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1), GR(z0, 0), P(z0))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)), GR(z1, 0), P(z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
GR(0, z0) → c7
GR(s(z0), 0) → c8
GR(s(z0), s(z1)) → c9(GR(z0, z1))
AND(true, true) → c10
AND(false, z0) → c11
AND(z0, false) → c12
P(0) → c13
P(s(z0)) → c14

K tuples:none
Defined Rule Symbols:

cond1, cond2, cond3, cond4, gr, and, p

Defined Pair Symbols:

COND1, COND2, COND3, COND4, GR, AND, P

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 7 trailing nodes:

GR(0, z0) → c7
AND(false, z0) → c11
P(0) → c13
AND(z0, false) → c12
P(s(z0)) → c14
AND(true, true) → c10
GR(s(z0), 0) → c8

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1) → cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) → cond3(gr(z0, 0), z0, z1)
cond2(false, z0, z1) → cond4(gr(z1, 0), z0, z1)
cond3(true, z0, z1) → cond3(gr(z0, 0), p(z0), z1)
cond3(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
cond4(true, z0, z1) → cond4(gr(z1, 0), z0, p(z1))
cond4(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
p(0) → 0
p(s(z0)) → z0

Tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1), GR(z0, 0))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1), GR(z1, 0))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1), GR(z0, 0), P(z0))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)), GR(z1, 0), P(z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
GR(s(z0), s(z1)) → c9(GR(z0, z1))

S tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1), GR(z0, 0))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1), GR(z1, 0))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1), GR(z0, 0), P(z0))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)), GR(z1, 0), P(z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
GR(s(z0), s(z1)) → c9(GR(z0, z1))

K tuples:none
Defined Rule Symbols:

cond1, cond2, cond3, cond4, gr, and, p

Defined Pair Symbols:

COND1, COND2, COND3, COND4, GR

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c9

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

Removed 12 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1) → cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) → cond3(gr(z0, 0), z0, z1)
cond2(false, z0, z1) → cond4(gr(z1, 0), z0, z1)
cond3(true, z0, z1) → cond3(gr(z0, 0), p(z0), z1)
cond3(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
cond4(true, z0, z1) → cond4(gr(z1, 0), z0, p(z1))
cond4(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
p(0) → 0
p(s(z0)) → z0

Tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))

S tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))

K tuples:none
Defined Rule Symbols:

cond1, cond2, cond3, cond4, gr, and, p

Defined Pair Symbols:

COND1, GR, COND2, COND3, COND4

Compound Symbols:

c, c9, c1, c2, c3, c4, c5, c6

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

cond1(true, z0, z1) → cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) → cond3(gr(z0, 0), z0, z1)
cond2(false, z0, z1) → cond4(gr(z1, 0), z0, z1)
cond3(true, z0, z1) → cond3(gr(z0, 0), p(z0), z1)
cond3(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
cond4(true, z0, z1) → cond4(gr(z1, 0), z0, p(z1))
cond4(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))

S tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))

K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

COND1, GR, COND2, COND3, COND4

Compound Symbols:

c, c9, c1, c2, c3, c4, c5, c6

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

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

COND1(true, 0, z0) → c(COND2(false, 0, z0), GR(0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0), GR(s(z0), 0))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0), GR(0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0), GR(s(z0), 0))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0), GR(0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0), GR(s(z0), 0))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND2, COND3, COND4, COND1

Compound Symbols:

c9, c1, c2, c3, c4, c5, c6, c

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

Removed 2 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))

K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND2, COND3, COND4, COND1

Compound Symbols:

c9, c1, c2, c3, c4, c5, c6, c, c

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

Use narrowing to replace COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1)) by

COND2(true, 0, x1) → c1(COND3(false, 0, x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, 0, x1) → c1(COND3(false, 0, x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, 0, x1) → c1(COND3(false, 0, x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND2, COND3, COND4, COND1

Compound Symbols:

c9, c2, c3, c4, c5, c6, c, c, c1

(15) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

COND2(true, 0, x1) → c1(COND3(false, 0, x1))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND2, COND3, COND4, COND1

Compound Symbols:

c9, c2, c3, c4, c5, c6, c, c, c1

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

Use narrowing to replace COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1)) by

COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))

K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND3, COND4, COND1, COND2

Compound Symbols:

c9, c3, c4, c5, c6, c, c, c1, c2

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

Use narrowing to replace COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1)) by

COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))

K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND3, COND4, COND1, COND2

Compound Symbols:

c9, c4, c5, c6, c, c, c1, c2, c3

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

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

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))

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

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₂ + x₃
POL(COND2(x₁, x₂, x₃)) = x₂ + x₃
POL(COND3(x₁, x₂, x₃)) = x₂ + x₃
POL(COND4(x₁, x₂, x₃)) = x₂ + x₃
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = [1]
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(p(x₁)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND3, COND4, COND1, COND2

Compound Symbols:

c9, c4, c5, c6, c, c, c1, c2, c3

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

Use narrowing to replace COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1)) by

COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND4, COND1, COND2, COND3

Compound Symbols:

c9, c5, c6, c, c, c1, c2, c3, c4

(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.

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))

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)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

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(COND3(x₁, x₂, x₃)) = [1]
POL(COND4(x₁, x₂, x₃)) = [1]
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = x₁
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(gr(x₁, x₂)) = [1]
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
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND4, COND1, COND2, COND3

Compound Symbols:

c9, c5, c6, c, c, c1, c2, c3, c4

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

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

COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))

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)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

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(COND3(x₁, x₂, x₃)) = [1]
POL(COND4(x₁, x₂, x₃)) = [1]
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = x₂
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(gr(x₁, x₂)) = [1]
POL(p(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = [1]

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND4, COND1, COND2, COND3

Compound Symbols:

c9, c5, c6, c, c, c1, c2, c3, c4

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

Use narrowing to replace COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1))) by

COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND4, COND1, COND2, COND3

Compound Symbols:

c9, c6, c, c, c1, c2, c3, c4, c5

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

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

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(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)) = x₃
POL(COND2(x₁, x₂, x₃)) = x₃
POL(COND3(x₁, x₂, x₃)) = x₃
POL(COND4(x₁, x₂, x₃)) = x₃
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(p(x₁)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND4, COND1, COND2, COND3

Compound Symbols:

c9, c6, c, c, c1, c2, c3, c4, c5

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

Use narrowing to replace COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1)) by

COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c, c1, c2, c3, c4, c5, c6

(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.

COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))

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)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

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(COND3(x₁, x₂, x₃)) = [1]
POL(COND4(x₁, x₂, x₃)) = [1]
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = x₂
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(gr(x₁, x₂)) = [1]
POL(p(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = [1]

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c, c1, c2, c3, c4, c5, c6

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

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

COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))

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)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

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

POL(0) = [1]
POL(COND1(x₁, x₂, x₃)) = x₁
POL(COND2(x₁, x₂, x₃)) = [1]
POL(COND3(x₁, x₂, x₃)) = [1]
POL(COND4(x₁, x₂, x₃)) = [1]
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = x₁
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(gr(x₁, x₂)) = x₂
POL(p(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = [1]

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c, c1, c2, c3, c4, c5, c6

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

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

COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c3, c4, c5, c6, c

(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.

COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

We considered the (Usable) Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)

And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

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

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [1]
POL(COND2(x₁, x₂, x₃)) = x₁
POL(COND3(x₁, x₂, x₃)) = [1]
POL(COND4(x₁, x₂, x₃)) = [1]
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = [1]
POL(gr(x₁, x₂)) = [1]
POL(p(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = [1]

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c3, c4, c5, c6, c

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

Use narrowing to replace COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1)) by

COND3(true, 0, x0) → c3(COND3(false, 0, x0))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c3, c4, c5, c6, c

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

Use narrowing to replace COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1)) by

COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c3, c4, c5, c6, c

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

Use narrowing to replace COND3(true, 0, x1) → c3(COND3(false, p(0), x1)) by

COND3(true, 0, x0) → c3(COND3(false, 0, x0))

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c3, c4, c5, c6, c

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

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

COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c3, c4, c5, c6, c

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

Use narrowing to replace COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1)) by

COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c4, c5, c6, c, c3

(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.

COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

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

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

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(COND3(x₁, x₂, x₃)) = x₂
POL(COND4(x₁, x₂, x₃)) = x₂
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(p(x₁)) = 0
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c4, c5, c6, c, c3

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

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

COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c4, c5, c6, c, c3

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

Use narrowing to replace COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0)) by

COND3(false, x0, 0) → c4(COND1(false, x0, 0))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, x0, 0) → c4(COND1(false, x0, 0))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c4, c5, c6, c, c3

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

Removed 1 trailing nodes:

COND3(false, x0, 0) → c4(COND1(false, x0, 0))

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c4, c5, c6, c, c3

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

Use narrowing to replace COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0))) by

COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c4, c5, c6, c, c3

(63) 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, 0, z0) → c(COND2(false, 0, 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)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))

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)) = x₂
POL(COND3(x₁, x₂, x₃)) = [1]
POL(COND4(x₁, x₂, x₃)) = x₂
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = x₁
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(gr(x₁, x₂)) = x₁
POL(p(x₁)) = 0
POL(s(x₁)) = [1]
POL(true) = [1]

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c4, c5, c6, c, c3

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

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

COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c4, c5, c6, c, c3

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

Use narrowing to replace COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1)) by

COND3(false, 0, x0) → c4(COND1(false, 0, x0))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, 0, s(z0)) → c4(COND1(and(false, true), 0, s(z0)))

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND3(false, 0, x0) → c4(COND1(false, 0, x0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c4, c5, c6, c, c3

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

Removed 1 trailing nodes:

COND3(false, 0, x0) → c4(COND1(false, 0, x0))

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c4, c5, c6, c, c3

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

Use narrowing to replace COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1)) by

COND3(false, s(x0), 0) → c4(COND1(and(true, false), s(x0), 0))
COND3(false, s(x0), s(z0)) → c4(COND1(and(true, true), s(x0), s(z0)))

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c5, c6, c, c3, c4

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

Use narrowing to replace COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0)) by

COND4(true, x0, 0) → c5(COND4(false, x0, 0))

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c5, c6, c, c3, c4

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

Use narrowing to replace COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0)) by

COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c5, c6, c, c3, c4

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

Use narrowing to replace COND4(true, x0, 0) → c5(COND4(false, x0, p(0))) by

COND4(true, x0, 0) → c5(COND4(false, x0, 0))

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c5, c6, c, c3, c4

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

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

COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))

Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c5, c6, c, c3, c4

(81) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

p(0) → 0

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))

Defined Rule Symbols:

p, and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c5, c6, c, c3, c4

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

Use narrowing to replace COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0)))) by

COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))

Defined Rule Symbols:

p, and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

(85) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

p(s(z0)) → z0

(86) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

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

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

COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

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

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(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)) = x₃
POL(COND3(x₁, x₂, x₃)) = x₃
POL(COND4(x₁, x₂, x₃)) = x₃
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(88) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

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

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

COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

(90) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

(91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

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

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

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

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [2]x₂²
POL(COND2(x₁, x₂, x₃)) = [2]x₂²
POL(COND3(x₁, x₂, x₃)) = [2]x₁·x₂
POL(COND4(x₁, x₂, x₃)) = [2]x₂²
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = [2] + x₂
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = [2]
POL(gr(x₁, x₂)) = 0
POL(s(x₁)) = [1]
POL(true) = 0

(92) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

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

Use narrowing to replace COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0)) by

COND4(false, x0, 0) → c6(COND1(false, x0, 0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))

(94) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, x0, 0) → c6(COND1(false, x0, 0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

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

Removed 1 trailing nodes:

COND4(false, x0, 0) → c6(COND1(false, x0, 0))

(96) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

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

Use narrowing to replace COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0))) by

COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

(98) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

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

Use narrowing to replace COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1)) by

COND4(false, 0, x0) → c6(COND1(false, 0, x0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, 0, s(z0)) → c6(COND1(and(false, true), 0, s(z0)))

(100) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))
COND4(false, 0, x0) → c6(COND1(false, 0, x0))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

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

Removed 1 trailing nodes:

COND4(false, 0, x0) → c6(COND1(false, 0, x0))

(102) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

(103) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))

We considered the (Usable) Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)

And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

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

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [2]x₂·x₃
POL(COND2(x₁, x₂, x₃)) = [2]x₁·x₂
POL(COND3(x₁, x₂, x₃)) = [2]x₁·x₂
POL(COND4(x₁, x₂, x₃)) = [2]x₂ + [2]x₁·x₃
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = [2]
POL(gr(x₁, x₂)) = [2]
POL(s(x₁)) = [2]
POL(true) = 0

(104) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

(105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

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

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

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

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₂·x₃
POL(COND2(x₁, x₂, x₃)) = 0
POL(COND3(x₁, x₂, x₃)) = x₁·x₂
POL(COND4(x₁, x₂, x₃)) = x₁·x₃
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = [2]
POL(gr(x₁, x₂)) = 0
POL(s(x₁)) = [2]
POL(true) = 0

(106) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))

K tuples:

COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

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

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

COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

(108) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))

K tuples:

COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

(109) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

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

We considered the (Usable) Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)

And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

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

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = [1] + [2]x₂·x₃ + [2]x₂²
POL(COND2(x₁, x₂, x₃)) = [2]x₂ + [2]x₂·x₃ + x₁²
POL(COND3(x₁, x₂, x₃)) = [1] + [2]x₃ + [2]x₁·x₂
POL(COND4(x₁, x₂, x₃)) = [1] + [2]x₂ + [2]x₁·x₃
POL(GR(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = [1]
POL(gr(x₁, x₂)) = [1]
POL(s(x₁)) = [1]
POL(true) = 0

(110) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))

K tuples:

COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

c9, c, c1, c2, c6, c, c3, c4, c5

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

Use narrowing to replace COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1)) by

COND4(false, s(x0), 0) → c6(COND1(and(true, false), s(x0), 0))
COND4(false, s(x0), s(z0)) → c6(COND1(and(true, true), s(x0), s(z0)))

(112) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))

K tuples:

COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c, c3, c4, c5, c6

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

Split RHS of tuples not part of any SCC

(114) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))
COND1(true, s(0), s(z0)) → c7(COND2(false, s(0), s(z0)))
COND1(true, s(0), s(z0)) → c7(GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c7(COND2(true, s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(0)) → c7(GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c7(COND2(gr(z0, z1), s(s(z0)), s(s(z1))))
COND1(true, s(s(z0)), s(s(z1))) → c7(GR(s(s(z0)), s(s(z1))))

S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))

K tuples:

COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND1(true, s(0), s(z0)) → c7(COND2(false, s(0), s(z0)))
COND1(true, s(0), s(z0)) → c7(GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c7(COND2(true, s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(0)) → c7(GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c7(COND2(gr(z0, z1), s(s(z0)), s(s(z1))))
COND1(true, s(s(z0)), s(s(z1))) → c7(GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c3, c4, c5, c6, c7

(115) 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)) → c9(GR(z0, z1))

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

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))
COND1(true, s(0), s(z0)) → c7(COND2(false, s(0), s(z0)))
COND1(true, s(0), s(z0)) → c7(GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c7(COND2(true, s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(0)) → c7(GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c7(COND2(gr(z0, z1), s(s(z0)), s(s(z1))))
COND1(true, s(s(z0)), s(s(z1))) → c7(GR(s(s(z0)), s(s(z1))))

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

POL(0) = [1]
POL(COND1(x₁, x₂, x₃)) = [1] + x₂ + x₃
POL(COND2(x₁, x₂, x₃)) = [1] + x₂ + x₃
POL(COND3(x₁, x₂, x₃)) = [1] + x₂ + x₃
POL(COND4(x₁, x₂, x₃)) = [1] + x₂ + x₃
POL(GR(x₁, x₂)) = x₁ + x₂
POL(and(x₁, x₂)) = [1] + x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(gr(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = [1] + x₁
POL(true) = [1]

(116) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(false, 0, 0) → c6(COND1(and(false, false), 0, 0))
COND4(false, s(z0), 0) → c6(COND1(and(true, false), s(z0), 0))
COND4(false, 0, s(x1)) → c6(COND1(and(false, true), 0, s(x1)))
COND4(false, s(z0), s(x1)) → c6(COND1(and(true, true), s(z0), s(x1)))
COND1(true, s(0), s(z0)) → c7(COND2(false, s(0), s(z0)))
COND1(true, s(0), s(z0)) → c7(GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c7(COND2(true, s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(0)) → c7(GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c7(COND2(gr(z0, z1), s(s(z0)), s(s(z1))))
COND1(true, s(s(z0)), s(s(z1))) → c7(GR(s(s(z0)), s(s(z1))))

S tuples:none
K tuples:

COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND1(true, s(0), s(z0)) → c7(COND2(false, s(0), s(z0)))
COND1(true, s(0), s(z0)) → c7(GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c7(COND2(true, s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(0)) → c7(GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c7(COND2(gr(z0, z1), s(s(z0)), s(s(z1))))
COND1(true, s(s(z0)), s(s(z1))) → c7(GR(s(s(z0)), s(s(z1))))
GR(s(z0), s(z1)) → c9(GR(z0, z1))

Defined Rule Symbols:

and, gr

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

c9, c, c1, c2, c3, c4, c5, c6, c7

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

The set S is empty