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 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CdtProblem
↳15 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CdtProblem
↳17 CdtUsableRulesProof (⇔, 0 ms)
↳18 CdtProblem
↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 45 ms)
↳20 CdtProblem
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳22 CdtProblem
↳23 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳24 CdtProblem
↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳26 CdtProblem
↳27 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳28 CdtProblem
↳29 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳30 CdtProblem
↳31 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)
↳32 CdtProblem
↳33 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 37 ms)
↳34 CdtProblem
↳35 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 19 ms)
↳36 CdtProblem
↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳38 CdtProblem
↳39 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳40 CdtProblem
↳41 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳42 CdtProblem
↳43 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳44 CdtProblem
↳45 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳46 CdtProblem
↳47 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳48 CdtProblem
↳49 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳50 CdtProblem
↳51 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳52 CdtProblem
↳53 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳54 CdtProblem
↳55 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳56 CdtProblem
↳57 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳58 CdtProblem
↳59 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳60 CdtProblem
↳61 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳62 CdtProblem
↳63 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)
↳64 CdtProblem
↳65 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
↳66 CdtProblem
↳67 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
↳68 CdtProblem
↳69 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳70 CdtProblem
↳71 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳72 CdtProblem
↳73 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳74 CdtProblem
↳75 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳76 CdtProblem
↳77 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳78 CdtProblem
↳79 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 7 ms)
↳80 CdtProblem
↳81 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 117 ms)
↳82 CdtProblem
↳83 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳84 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:

cond(true, x, y) → cond(gr(x, y), p(x), s(y))
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1))
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1), P(z0))
GR(0, z0) → c1
GR(s(z0), 0) → c2
GR(s(z0), s(z1)) → c3(GR(z0, z1))
P(0) → c4
P(s(z0)) → c5
S tuples:

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1), P(z0))
GR(0, z0) → c1
GR(s(z0), 0) → c2
GR(s(z0), s(z1)) → c3(GR(z0, z1))
P(0) → c4
P(s(z0)) → c5
K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR, P

Compound Symbols:

c, c1, c2, c3, c4, c5

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

Removed 4 trailing nodes:

P(s(z0)) → c5
P(0) → c4
GR(0, z0) → c1
GR(s(z0), 0) → c2

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1))
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1), P(z0))
GR(s(z0), s(z1)) → c3(GR(z0, z1))
S tuples:

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1), P(z0))
GR(s(z0), s(z1)) → c3(GR(z0, z1))
K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c3

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1))
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1))
K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(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
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1))
K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

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

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

COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(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
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0))

(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
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

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

Removed 2 trailing tuple parts

(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
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

(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
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

(17) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

p(0) → 0

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
We considered the (Usable) Rules:

p(s(z0)) → z0
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(COND(x1, x2, x3)) = x2   
POL(GR(x1, x2)) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = 0   
POL(gr(x1, x2)) = 0   
POL(p(x1)) = x1   
POL(s(x1)) = [1] + x1   
POL(true) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Use narrowing to replace COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) by

COND(true, s(z0), 0) → c(COND(true, z0, s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), 0) → c(COND(true, z0, s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

(23) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

COND(true, s(z0), 0) → c(COND(true, z0, s(0)), GR(s(z0), 0))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(s(z0))), GR(s(0), s(z0)))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(s(z0))), GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(s(z0))), GR(s(0), s(z0)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Split RHS of tuples not part of any SCC

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

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

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

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

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(COND(x1, x2, x3)) = [3] + [3]x3   
POL(GR(x1, x2)) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1(x1)) = x1   
POL(c3(x1)) = x1   
POL(false) = [1]   
POL(gr(x1, x2)) = [2]   
POL(p(x1)) = 0   
POL(s(x1)) = 0   
POL(true) = [1]   

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

(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.

COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
We considered the (Usable) Rules:

p(s(z0)) → z0
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(COND(x1, x2, x3)) = [2]x2   
POL(GR(x1, x2)) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1(x1)) = x1   
POL(c3(x1)) = x1   
POL(false) = 0   
POL(gr(x1, x2)) = 0   
POL(p(x1)) = x1   
POL(s(x1)) = [2] + x1   
POL(true) = 0   

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Use narrowing to replace COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) by

COND(true, 0, z0) → c(COND(false, 0, s(z0)))

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, 0, z0) → c(COND(false, 0, s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, 0, z0) → c(COND(false, 0, s(z0)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c(COND(false, 0, s(z0)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

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

COND(true, s(z0), 0) → c(COND(true, z0, s(0)))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(z0), 0) → c(COND(true, z0, s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

(43) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

COND(true, s(z0), 0) → c(COND(true, z0, s(0)))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

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

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

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(COND(false, 0, s(s(z0))), GR(s(0), s(z0)))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

(47) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(s(0))), GR(s(s(z0)), s(0)))

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(COND(false, 0, s(s(z0))), GR(s(0), s(z0)))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Removed 1 trailing tuple parts

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Use narrowing to replace COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1))) by

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

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(COND(false, 0, s(s(z0))), GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

(53) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(s(0))), GR(s(s(z0)), s(0)))

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(0), s(z0)) → c(COND(false, 0, s(s(z0))), GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Removed 1 trailing tuple parts

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

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

COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(COND(false, p(s(s(0))), s(s(s(z0)))), GR(s(s(0)), s(s(z0))))

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(COND(false, p(s(s(0))), s(s(s(z0)))), GR(s(s(0)), s(s(z0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(COND(false, p(s(s(0))), s(s(s(z0)))), GR(s(s(0)), s(s(z0))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c

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

Removed 1 trailing tuple parts

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c

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

Split RHS of tuples not part of any SCC

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c, c2

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

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

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c, c2

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

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

COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(COND(x1, x2, x3)) = [1] + x3   
POL(GR(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(false) = 0   
POL(gr(x1, x2)) = 0   
POL(p(x1)) = 0   
POL(s(x1)) = [1]   
POL(true) = 0   

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c, c2

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

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
We considered the (Usable) Rules:

gr(0, z0) → false
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(COND(x1, x2, x3)) = x1 + x2   
POL(GR(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(false) = 0   
POL(gr(x1, x2)) = [1]   
POL(p(x1)) = x1   
POL(s(x1)) = [1] + x1   
POL(true) = [1]   

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c, c2

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

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

COND(true, s(s(x0)), s(0)) → c1(COND(true, s(x0), s(s(0))))

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(x0)), s(0)) → c1(COND(true, s(x0), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c2, c1

(71) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

COND(true, s(s(x0)), s(0)) → c1(COND(true, s(x0), s(s(0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c2

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

Use forward instantiation to replace GR(s(z0), s(z1)) → c3(GR(z0, z1)) by

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
S tuples:

COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

Removed 1 trailing nodes:

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
S tuples:

COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

Used rewriting to replace COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1))))) by COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
S tuples:

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

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

COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
We considered the (Usable) Rules:

p(s(z0)) → z0
And the Tuples:

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [2]   
POL(COND(x1, x2, x3)) = [2]x2   
POL(GR(x1, x2)) = [2]   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(false) = 0   
POL(gr(x1, x2)) = 0   
POL(p(x1)) = x1   
POL(s(x1)) = [2] + x1   
POL(true) = 0   

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
S tuples:

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

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

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
We considered the (Usable) Rules:

p(s(z0)) → z0
And the Tuples:

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(COND(x1, x2, x3)) = [2]x22   
POL(GR(x1, x2)) = [2]x1   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(false) = [1]   
POL(gr(x1, x2)) = 0   
POL(p(x1)) = x1   
POL(s(x1)) = [2] + x1   
POL(true) = 0   

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
S tuples:none
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

The set S is empty

(84) BOUNDS(1, 1)