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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 0 ms)
↳8 CdtProblem
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 8 ms)
↳10 CdtProblem
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CdtProblem
↳15 CdtUsableRulesProof (⇔, 0 ms)
↳16 CdtProblem
↳17 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 72 ms)
↳18 CdtProblem
↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳20 CdtProblem
↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳22 CdtProblem
↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳24 CdtProblem
↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳26 CdtProblem
↳27 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)
↳28 CdtProblem
↳29 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 7 ms)
↳30 CdtProblem
↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳32 CdtProblem
↳33 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳34 CdtProblem
↳35 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)
↳36 CdtProblem
↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳38 CdtProblem
↳39 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳40 CdtProblem
↳41 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)
↳42 CdtProblem
↳43 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳44 CdtProblem
↳45 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳46 CdtProblem
↳47 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 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 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 3 ms)
↳54 CdtProblem
↳55 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳56 CdtProblem
↳57 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳58 CdtProblem
↳59 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)
↳60 CdtProblem
↳61 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)
↳62 CdtProblem
↳63 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳64 CdtProblem
↳65 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳66 CdtProblem
↳67 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳68 CdtProblem
↳69 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳70 CdtProblem
↳71 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳72 CdtProblem
↳73 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳74 CdtProblem
↳75 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳76 CdtProblem
↳77 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳78 CdtProblem
↳79 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳80 CdtProblem
↳81 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)
↳82 CdtProblem
↳83 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳84 CdtProblem
↳85 CdtUsableRulesProof (⇔, 0 ms)
↳86 CdtProblem
↳87 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)
↳88 CdtProblem
↳89 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 24 ms)
↳90 CdtProblem
↳91 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳92 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

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), 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), 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), 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:

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), 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), 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), 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), 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), 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), 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), 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), 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), z1), GR(z0, z1)) by

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

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

(15) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

p(0) → 0

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, 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, x1), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 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) = [2]   
POL(gr(x1, x2)) = [5] + [2]x2   
POL(p(x1)) = x1   
POL(s(x1)) = [1] + x1   
POL(true) = [3]   

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

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

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

(23) 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(z1)), GR(s(z0), s(z1))) by

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

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

(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, x1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1)))
COND(true, s(z0), 0) → c(COND(true, z0, 0))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), 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(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, x1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(x1)), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), 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(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), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

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

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

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

COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1))
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

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

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

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

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

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

COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1))
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(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

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

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

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

COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1))
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(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

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

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

COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1))
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(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

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

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

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

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

POL(0) = [4]   
POL(COND(x1, x2, x3)) = [4] + [3]x3   
POL(GR(x1, x2)) = 0   
POL(c) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [3]   
POL(gr(x1, x2)) = [4] + [5]x1 + x2   
POL(p(x1)) = [4]x1   
POL(s(x1)) = [2] + x1   
POL(true) = [1]   

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

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

COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1))
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(z1)), GR(s(z0), s(z1)))
COND(true, 0, z0) → c
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

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

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

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

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

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

COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1))
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(z1)), GR(s(z0), s(z1)))
COND(true, 0, z0) → c
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c

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

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(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

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

POL(0) = [4]   
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) = [3]   
POL(gr(x1, x2)) = [4] + [2]x1   
POL(p(x1)) = x1   
POL(s(x1)) = [4] + x1   
POL(true) = [3]   

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

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

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

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

(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), 0) → c(COND(true, z0, 0))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), 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(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)) → c(COND(true, s(z0), 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(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(z0)), GR(s(0), s(z0)))
S tuples:

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(s(x0)), s(0)) → c(COND(true, s(x0), s(0)), GR(s(s(x0)), s(0)))

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
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), 0) → c(COND(true, z0, 0))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), 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)) → c(COND(true, s(z0), 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(z1))), GR(s(s(z0)), s(s(z1))))
COND(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) = [2]   
POL(COND(x1, x2, x3)) = [2] + x2 + [4]x3   
POL(GR(x1, x2)) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [2]   
POL(gr(x1, x2)) = [4]x1 + [2]x2   
POL(p(x1)) = x1   
POL(s(x1)) = [1] + x1   
POL(true) = 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(z0), 0) → c(COND(true, z0, 0))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), 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)) → c(COND(true, s(z0), 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(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(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

(55) 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(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(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(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(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(z0))), GR(s(s(0)), s(s(z0))))

(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(z0), 0) → c(COND(true, z0, 0))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), 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(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(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(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(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(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(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(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(z0))), GR(s(s(0)), s(s(z0))))
K tuples:

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

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

POL(0) = [4]   
POL(COND(x1, x2, x3)) = [4]x3   
POL(GR(x1, x2)) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [3]   
POL(gr(x1, x2)) = [4] + [5]x1 + x2   
POL(p(x1)) = [3] + [4]x1   
POL(s(x1)) = [5] + x1   
POL(true) = [4]   

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

(61) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

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

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

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

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Use instantiation to replace COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) by

COND(true, s(0), s(0)) → c(GR(s(0), s(0)))
COND(true, s(0), s(s(x1))) → c(GR(s(0), s(s(x1))))
COND(true, s(0), s(s(0))) → c(GR(s(0), s(s(0))))
COND(true, s(0), s(s(s(x1)))) → c(GR(s(0), s(s(s(x1)))))

(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(z0), 0) → c(COND(true, z0, 0))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), 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(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(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(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(0), s(0)) → c(GR(s(0), s(0)))
COND(true, s(0), s(s(x1))) → c(GR(s(0), s(s(x1))))
COND(true, s(0), s(s(0))) → c(GR(s(0), s(s(0))))
COND(true, s(0), s(s(s(x1)))) → c(GR(s(0), s(s(s(x1)))))
S tuples:

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

COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1)))
COND(true, s(z0), 0) → c(COND(true, z0, 0))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), 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(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(0)) → c(GR(s(0), s(0)))
COND(true, s(0), s(s(x1))) → c(GR(s(0), s(s(x1))))
COND(true, s(0), s(s(0))) → c(GR(s(0), s(s(0))))
COND(true, s(0), s(s(s(x1)))) → c(GR(s(0), s(s(s(x1)))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

(65) 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)))

(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:

COND(true, s(z0), 0) → c(COND(true, z0, 0))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), 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(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(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(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(0), s(0)) → c(GR(s(0), s(0)))
COND(true, s(0), s(s(x1))) → c(GR(s(0), s(s(x1))))
COND(true, s(0), s(s(0))) → c(GR(s(0), s(s(0))))
COND(true, s(0), s(s(s(x1)))) → c(GR(s(0), s(s(s(x1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
S tuples:

COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), 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(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(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1)))
COND(true, s(z0), 0) → c(COND(true, z0, 0))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), 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(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(0)) → c(GR(s(0), s(0)))
COND(true, s(0), s(s(x1))) → c(GR(s(0), s(s(x1))))
COND(true, s(0), s(s(0))) → c(GR(s(0), s(s(0))))
COND(true, s(0), s(s(s(x1)))) → c(GR(s(0), s(s(s(x1)))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

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

Removed 5 trailing nodes:

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(0), s(s(x1))) → c(GR(s(0), s(s(x1))))
COND(true, s(0), s(0)) → c(GR(s(0), s(0)))
COND(true, s(0), s(s(0))) → c(GR(s(0), s(s(0))))
COND(true, s(0), s(s(s(x1)))) → c(GR(s(0), s(s(s(x1)))))

(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:

COND(true, s(z0), 0) → c(COND(true, z0, 0))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), 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(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(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(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))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
S tuples:

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

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

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

Removed 3 trailing tuple parts

(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:

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

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

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

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

Use forward instantiation to replace COND(true, s(z0), 0) → c(COND(true, z0, 0)) by

COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 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:

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

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

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

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

Use forward instantiation to replace COND(true, s(z0), 0) → c(COND(true, z0, 0)) by

COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(y0))), 0) → c(COND(true, s(s(y0)), 0))

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

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

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

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

Use forward instantiation to replace COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1))) by

COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))

(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(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(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(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))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(y0))), 0) → c(COND(true, s(s(y0)), 0))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
S tuples:

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

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

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

Use forward instantiation to replace COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1))) by

COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))

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

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

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

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

Used rewriting to replace COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0)))) by COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(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(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), 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))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(y0))), 0) → c(COND(true, s(s(y0)), 0))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
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(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
K tuples:

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(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

(81) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

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

POL(0) = [4]   
POL(COND(x1, x2, x3)) = [3] + [2]x2   
POL(GR(x1, x2)) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [3]   
POL(gr(x1, x2)) = x2   
POL(p(x1)) = x1   
POL(s(x1)) = [1] + 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(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), 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))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(y0))), 0) → c(COND(true, s(s(y0)), 0))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
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(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(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(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

(83) 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(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(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))

(84) 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(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(y0))), 0) → c(COND(true, s(s(y0)), 0))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), 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), s(s(z0)), 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(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

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(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

(85) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

p(s(z0)) → z0

(86) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(y0))), 0) → c(COND(true, s(s(y0)), 0))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), 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), s(s(z0)), 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(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

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(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
Defined Rule Symbols:

gr

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

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

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

COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
We considered the (Usable) Rules:none
And the Tuples:

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(y0))), 0) → c(COND(true, s(s(y0)), 0))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), 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), s(s(z0)), 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) = [4]   
POL(COND(x1, x2, x3)) = [4]x2   
POL(GR(x1, x2)) = [5]   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [3]   
POL(gr(x1, x2)) = [4] + [5]x1 + x2   
POL(s(x1)) = [5] + x1   
POL(true) = [4]   

(88) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

gr

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

(89) CdtRuleRemovalProof (UPPER BOUND (ADD(O(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:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(y0))), 0) → c(COND(true, s(s(y0)), 0))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), 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), s(s(z0)), 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]x2 + x2·x3 + [2]x1·x2   
POL(GR(x1, x2)) = [1] + [2]x2   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [2]   
POL(gr(x1, x2)) = [2]   
POL(s(x1)) = [2] + x1   
POL(true) = [2]   

(90) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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(z1))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
COND(true, s(s(y0)), 0) → c(COND(true, s(y0), 0))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, s(s(z0)), 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), s(s(z0)), 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

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c3

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

The set S is empty

(92) BOUNDS(O(1), O(1))