YES(O(1), O(n^2)) 98.98/44.77 YES(O(1), O(n^2)) 98.98/44.79 98.98/44.79 98.98/44.79 98.98/44.79 98.98/44.79 98.98/44.79 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 98.98/44.79 98.98/44.79 98.98/44.79
98.98/44.79
98.98/44.79
98.98/44.79
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
98.98/44.79 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳2 CdtProblem
98.98/44.79 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳4 CdtProblem
98.98/44.79 
↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳6 CdtProblem
98.98/44.79 
↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳8 CdtProblem
98.98/44.79 
↳9 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳10 CdtProblem
98.98/44.79 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
98.98/44.79 
↳12 CdtProblem
98.98/44.79 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
98.98/44.79 
↳14 CdtProblem
98.98/44.79 
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳16 CdtProblem
98.98/44.79 
↳17 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳18 CdtProblem
98.98/44.79 
↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳20 CdtProblem
98.98/44.79 
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳22 CdtProblem
98.98/44.79 
↳23 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳24 CdtProblem
98.98/44.79 
↳25 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳26 CdtProblem
98.98/44.79 
↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
98.98/44.79 
↳28 CdtProblem
98.98/44.79 
↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
98.98/44.79 
↳30 CdtProblem
98.98/44.79 
↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳32 CdtProblem
98.98/44.79 
↳33 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳34 CdtProblem
98.98/44.79 
↳35 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳36 CdtProblem
98.98/44.79 
↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳38 CdtProblem
98.98/44.79 
↳39 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳40 CdtProblem
98.98/44.79 
↳41 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
98.98/44.79 
↳42 CdtProblem
98.98/44.79 
↳43 CdtRewritingProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳44 CdtProblem
98.98/44.79 
↳45 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳46 CdtProblem
98.98/44.79 
↳47 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
98.98/44.79 
↳48 CdtProblem
98.98/44.79 
↳49 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳50 CdtProblem
98.98/44.79 
↳51 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳52 CdtProblem
98.98/44.79 
↳53 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳54 CdtProblem
98.98/44.79 
↳55 CdtRewritingProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳56 CdtProblem
98.98/44.79 
↳57 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳58 CdtProblem
98.98/44.79 
↳59 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
98.98/44.79 
↳60 CdtProblem
98.98/44.79 
↳61 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
98.98/44.79 
↳62 CdtProblem
98.98/44.79 
↳63 SIsEmptyProof (BOTH BOUNDS(ID, ID))
98.98/44.79 
↳64 BOUNDS(O(1), O(1))
98.98/44.79
98.98/44.79 98.98/44.79
98.98/44.79
98.98/44.79

(0) Obligation:

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

cond(true, x, y) → cond(gr(x, y), p(x), y) 98.98/44.79
gr(0, x) → false 98.98/44.80
gr(s(x), 0) → true 98.98/44.80
gr(s(x), s(y)) → gr(x, y) 98.98/44.80
p(0) → 0 98.98/44.80
p(s(x)) → x

Rewrite Strategy: INNERMOST
98.98/44.80
98.98/44.80

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

Converted CpxTRS to CDT
98.98/44.80
98.98/44.80

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 98.98/44.80
gr(0, z0) → false 98.98/44.80
gr(s(z0), 0) → true 98.98/44.80
gr(s(z0), s(z1)) → gr(z0, z1) 98.98/44.80
p(0) → 0 98.98/44.80
p(s(z0)) → z0
Tuples:

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), z1), GR(z0, z1), P(z0)) 98.98/44.80
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)) 98.98/44.80
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

98.98/44.80
98.98/44.80

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

Removed 1 trailing tuple parts
98.98/44.80
98.98/44.80

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 98.98/44.80
gr(0, z0) → false 98.98/44.80
gr(s(z0), 0) → true 98.98/44.80
gr(s(z0), s(z1)) → gr(z0, z1) 98.98/44.80
p(0) → 0 98.98/44.80
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 98.98/44.80
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)) 98.98/44.80
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

99.34/44.81
99.34/44.81

(5) 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)) 99.34/44.81
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.81
COND(true, 0, z0) → c(COND(false, p(0), z0), GR(0, z0)) 99.34/44.81
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0), GR(s(z0), 0)) 99.34/44.81
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1)))
99.34/44.81
99.34/44.81

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.81
gr(0, z0) → false 99.34/44.81
gr(s(z0), 0) → true 99.34/44.81
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.81
p(0) → 0 99.34/44.81
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.81
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1), GR(0, x1)) 99.34/44.81
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.81
COND(true, 0, z0) → c(COND(false, p(0), z0), GR(0, z0)) 99.34/44.81
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0), GR(s(z0), 0)) 99.34/44.81
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)) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1), GR(0, x1)) 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.84
COND(true, 0, z0) → c(COND(false, p(0), z0), GR(0, z0)) 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0), GR(s(z0), 0)) 99.34/44.84
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:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

99.34/44.84
99.34/44.84

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

Removed 4 trailing tuple parts
99.34/44.84
99.34/44.84

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.84
gr(0, z0) → false 99.34/44.84
gr(s(z0), 0) → true 99.34/44.84
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.84
p(0) → 0 99.34/44.84
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, 0, z0) → c 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, 0, z0) → c 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.84
99.34/44.84

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c
99.34/44.84
99.34/44.84

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.84
gr(0, z0) → false 99.34/44.84
gr(s(z0), 0) → true 99.34/44.84
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.84
p(0) → 0 99.34/44.84
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, 0, z0) → c 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, 0, z0) → c 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.84
99.34/44.84

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

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

COND(true, 0, z0) → c
We considered the (Usable) Rules:

p(s(z0)) → z0 99.34/44.84
gr(0, z0) → false 99.34/44.84
gr(s(z0), 0) → true 99.34/44.84
gr(s(z0), s(z1)) → gr(z0, z1)
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, 0, z0) → c 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
The order we found is given by the following interpretation:
Polynomial interpretation : 99.34/44.84

POL(0) = 0    99.34/44.84
POL(COND(x1, x2, x3)) = [1]    99.34/44.84
POL(GR(x1, x2)) = 0    99.34/44.84
POL(c) = 0    99.34/44.84
POL(c(x1)) = x1    99.34/44.84
POL(c(x1, x2)) = x1 + x2    99.34/44.84
POL(c3(x1)) = x1    99.34/44.84
POL(false) = 0    99.34/44.84
POL(gr(x1, x2)) = 0    99.34/44.84
POL(p(x1)) = [1]    99.34/44.84
POL(s(x1)) = 0    99.34/44.84
POL(true) = 0   
99.34/44.84
99.34/44.84

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.84
gr(0, z0) → false 99.34/44.84
gr(s(z0), 0) → true 99.34/44.84
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.84
p(0) → 0 99.34/44.84
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, 0, z0) → c 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
K tuples:

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

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.84
99.34/44.84

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

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

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

p(s(z0)) → z0 99.34/44.84
gr(0, z0) → false 99.34/44.84
gr(s(z0), 0) → true 99.34/44.84
gr(s(z0), s(z1)) → gr(z0, z1)
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, 0, z0) → c 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
The order we found is given by the following interpretation:
Polynomial interpretation : 99.34/44.84

POL(0) = [2]    99.34/44.84
POL(COND(x1, x2, x3)) = x2    99.34/44.84
POL(GR(x1, x2)) = 0    99.34/44.84
POL(c) = 0    99.34/44.84
POL(c(x1)) = x1    99.34/44.84
POL(c(x1, x2)) = x1 + x2    99.34/44.84
POL(c3(x1)) = x1    99.34/44.84
POL(false) = [5]    99.34/44.84
POL(gr(x1, x2)) = [4] + [4]x1 + x2    99.34/44.84
POL(p(x1)) = x1    99.34/44.84
POL(s(x1)) = [1] + x1    99.34/44.84
POL(true) = [4]   
99.34/44.84
99.34/44.84

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.84
gr(0, z0) → false 99.34/44.84
gr(s(z0), 0) → true 99.34/44.84
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.84
p(0) → 0 99.34/44.84
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, 0, z0) → c 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
K tuples:

COND(true, 0, z0) → c 99.34/44.84
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.84
99.34/44.84

(15) 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)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1)))
99.34/44.84
99.34/44.84

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.84
gr(0, z0) → false 99.34/44.84
gr(s(z0), 0) → true 99.34/44.84
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.84
p(0) → 0 99.34/44.84
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.84
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.84
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.84
COND(true, 0, z0) → c 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.84
COND(true, s(z0), 0) → c(COND(true, z0, 0), GR(s(z0), 0)) 99.34/44.84
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)) 99.34/44.86
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.86
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
K tuples:

COND(true, 0, z0) → c 99.34/44.86
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.86
99.34/44.86

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

Removed 1 trailing tuple parts
99.34/44.86
99.34/44.86

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.86
gr(0, z0) → false 99.34/44.86
gr(s(z0), 0) → true 99.34/44.86
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.86
p(0) → 0 99.34/44.86
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.86
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.86
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.86
COND(true, 0, z0) → c 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.86
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, z0, 0))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.86
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.86
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
K tuples:

COND(true, 0, z0) → c 99.34/44.86
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, x1), GR(s(z0), x1))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.86
99.34/44.86

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c
99.34/44.86
99.34/44.86

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.86
gr(0, z0) → false 99.34/44.86
gr(s(z0), 0) → true 99.34/44.86
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.86
p(0) → 0 99.34/44.86
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.86
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.86
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.86
COND(true, 0, z0) → c 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.86
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, z0, 0))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.86
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(z1)), GR(s(z0), s(z1))) 99.34/44.86
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0))
K tuples:

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

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.86
99.34/44.86

(21) 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))) 99.34/44.86
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(z0)), GR(s(0), s(z0))) 99.34/44.86
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.86
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))))
99.34/44.86
99.34/44.86

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.86
gr(0, z0) → false 99.34/44.86
gr(s(z0), 0) → true 99.34/44.86
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.86
p(0) → 0 99.34/44.86
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.86
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.86
COND(true, 0, z0) → c 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.86
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.86
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(z0)), GR(s(0), s(z0))) 99.34/44.86
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.86
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))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.86
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.86
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(x1)), GR(s(z0), s(x1))) 99.34/44.86
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(z0)), GR(s(0), s(z0))) 99.34/44.86
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.86
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, 0, z0) → c
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.86
99.34/44.86

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

Removed 1 trailing tuple parts
99.34/44.86
99.34/44.86

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.86
gr(0, z0) → false 99.34/44.86
gr(s(z0), 0) → true 99.34/44.86
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.86
p(0) → 0 99.34/44.86
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.86
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.86
COND(true, 0, z0) → c 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.86
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.86
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.86
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)))) 99.34/44.86
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.86
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.86
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.86
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(x1)), GR(s(z0), s(x1))) 99.34/44.86
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.86
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)))) 99.34/44.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
K tuples:

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

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.87
99.34/44.87

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c
99.34/44.87
99.34/44.87

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.87
gr(0, z0) → false 99.34/44.87
gr(s(z0), 0) → true 99.34/44.87
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.87
p(0) → 0 99.34/44.87
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.87
COND(true, 0, z0) → c 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.87
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)))) 99.34/44.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.87
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(x1)), GR(s(z0), s(x1))) 99.34/44.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.87
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)))) 99.34/44.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
K tuples:

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

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.87
99.34/44.87

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

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

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
We considered the (Usable) Rules:

gr(0, z0) → false 99.34/44.87
gr(s(z0), 0) → true 99.34/44.87
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.87
p(s(z0)) → z0
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.87
COND(true, 0, z0) → c 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.87
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)))) 99.34/44.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 99.34/44.87

POL(0) = 0    99.34/44.87
POL(COND(x1, x2, x3)) = [4] + [5]x3    99.34/44.87
POL(GR(x1, x2)) = 0    99.34/44.87
POL(c) = 0    99.34/44.87
POL(c(x1)) = x1    99.34/44.87
POL(c(x1, x2)) = x1 + x2    99.34/44.87
POL(c3(x1)) = x1    99.34/44.87
POL(false) = 0    99.34/44.87
POL(gr(x1, x2)) = 0    99.34/44.87
POL(p(x1)) = [1]    99.34/44.87
POL(s(x1)) = [4]    99.34/44.87
POL(true) = 0   
99.34/44.87
99.34/44.87

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.87
gr(0, z0) → false 99.34/44.87
gr(s(z0), 0) → true 99.34/44.87
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.87
p(0) → 0 99.34/44.87
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.87
COND(true, 0, z0) → c 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.87
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)))) 99.34/44.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.87
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(x1)), GR(s(z0), s(x1))) 99.34/44.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.87
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, 0, z0) → c 99.34/44.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.87
99.34/44.87

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

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

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

gr(0, z0) → false 99.34/44.87
gr(s(z0), 0) → true 99.34/44.87
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.87
p(s(z0)) → z0
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.87
COND(true, 0, z0) → c 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.87
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)))) 99.34/44.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 99.34/44.87

POL(0) = 0    99.34/44.87
POL(COND(x1, x2, x3)) = x2    99.34/44.87
POL(GR(x1, x2)) = 0    99.34/44.87
POL(c) = 0    99.34/44.87
POL(c(x1)) = x1    99.34/44.87
POL(c(x1, x2)) = x1 + x2    99.34/44.87
POL(c3(x1)) = x1    99.34/44.87
POL(false) = 0    99.34/44.87
POL(gr(x1, x2)) = 0    99.34/44.87
POL(p(x1)) = x1    99.34/44.87
POL(s(x1)) = [2] + x1    99.34/44.87
POL(true) = 0   
99.34/44.87
99.34/44.87

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.87
gr(0, z0) → false 99.34/44.87
gr(s(z0), 0) → true 99.34/44.87
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.87
p(0) → 0 99.34/44.87
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.87
COND(true, 0, z0) → c 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.87
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.87
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)))) 99.34/44.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.88
COND(true, 0, x1) → c(COND(gr(0, x1), 0, x1)) 99.34/44.88
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.88
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.88
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, 0, z0) → c 99.34/44.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.88
99.34/44.88

(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))
99.34/44.88
99.34/44.88

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.88
gr(0, z0) → false 99.34/44.88
gr(s(z0), 0) → true 99.34/44.88
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.88
p(0) → 0 99.34/44.88
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.88
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, 0, z0) → c(COND(false, 0, z0))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, 0, z0) → c(COND(false, 0, z0))
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.89
99.34/44.89

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

Removed 1 trailing tuple parts
99.34/44.89
99.34/44.89

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, 0, z0) → c
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.89
99.34/44.89

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

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

COND(true, 0, z0) → c 99.34/44.89
COND(true, 0, z0) → c
99.34/44.89
99.34/44.89

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.89
99.34/44.89

(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))
99.34/44.89
99.34/44.89

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0))
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.89
99.34/44.89

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c
99.34/44.89
99.34/44.89

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0))
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.89
99.34/44.89

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

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

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

gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(s(z0)) → z0
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 99.34/44.89

POL(0) = 0    99.34/44.89
POL(COND(x1, x2, x3)) = x2 + x3    99.34/44.89
POL(GR(x1, x2)) = 0    99.34/44.89
POL(c) = 0    99.34/44.89
POL(c(x1)) = x1    99.34/44.89
POL(c(x1, x2)) = x1 + x2    99.34/44.89
POL(c3(x1)) = x1    99.34/44.89
POL(false) = 0    99.34/44.89
POL(gr(x1, x2)) = [2]x1    99.34/44.89
POL(p(x1)) = x1    99.34/44.89
POL(s(x1)) = [1] + x1    99.34/44.89
POL(true) = 0   
99.34/44.89
99.34/44.89

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
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, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.89
99.34/44.89

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

Used rewriting 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(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
99.34/44.89
99.34/44.89

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.89
99.34/44.89

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c
99.34/44.89
99.34/44.89

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.89
99.34/44.89

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

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

COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
We considered the (Usable) Rules:

gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(s(z0)) → z0
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 99.34/44.89

POL(0) = [1]    99.34/44.89
POL(COND(x1, x2, x3)) = [4]x2    99.34/44.89
POL(GR(x1, x2)) = 0    99.34/44.89
POL(c) = 0    99.34/44.89
POL(c(x1)) = x1    99.34/44.89
POL(c(x1, x2)) = x1 + x2    99.34/44.89
POL(c3(x1)) = x1    99.34/44.89
POL(false) = [3]    99.34/44.89
POL(gr(x1, x2)) = 0    99.34/44.89
POL(p(x1)) = x1    99.34/44.89
POL(s(x1)) = [4] + x1    99.34/44.89
POL(true) = 0   
99.34/44.89
99.34/44.89

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 99.34/44.89
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, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

99.34/44.89
99.34/44.89

(49) 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)))
99.34/44.89
99.34/44.89

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
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)))) 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0))) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
S tuples:

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)))) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)), GR(s(s(z0)), s(0)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c, c3

99.34/44.89
99.34/44.89

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

Removed 2 trailing tuple parts
99.34/44.89
99.34/44.89

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
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)))) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
S tuples:

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)))) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c, c3

99.34/44.89
99.34/44.89

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

Removed 2 trailing nodes:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c
99.34/44.89
99.34/44.89

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
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)))) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
S tuples:

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)))) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c, c3

99.34/44.89
99.34/44.89

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

Used rewriting 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(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
99.34/44.89
99.34/44.89

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0))) 99.34/44.89
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))))
S tuples:

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 99.34/44.89
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))))
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c, c3

99.34/44.89
99.34/44.89

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

Removed 2 trailing nodes:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(0), s(z0)) → c
99.34/44.89
99.34/44.89

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0))) 99.34/44.89
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))))
S tuples:

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 99.34/44.89
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))))
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c, c3

99.34/44.89
99.34/44.89

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

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

COND(true, s(s(z0)), 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:

gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1)
And the Tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0))) 99.34/44.89
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))))
The order we found is given by the following interpretation:
Polynomial interpretation : 99.34/44.89

POL(0) = [3]    99.34/44.89
POL(COND(x1, x2, x3)) = [4]x2    99.34/44.89
POL(GR(x1, x2)) = 0    99.34/44.89
POL(c) = 0    99.34/44.89
POL(c(x1)) = x1    99.34/44.89
POL(c(x1, x2)) = x1 + x2    99.34/44.89
POL(c3(x1)) = x1    99.34/44.89
POL(false) = [3]    99.34/44.89
POL(gr(x1, x2)) = 0    99.34/44.89
POL(s(x1)) = [4] + x1    99.34/44.89
POL(true) = 0   
99.34/44.89
99.34/44.89

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0))) 99.34/44.89
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))))
S tuples:

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

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0))) 99.34/44.89
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:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c, c3

99.34/44.89
99.34/44.89

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

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

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

gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1)
And the Tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0))) 99.34/44.89
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))))
The order we found is given by the following interpretation:
Polynomial interpretation : 99.34/44.89

POL(0) = 0    99.34/44.89
POL(COND(x1, x2, x3)) = x2·x3    99.34/44.89
POL(GR(x1, x2)) = x2    99.34/44.89
POL(c) = 0    99.34/44.89
POL(c(x1)) = x1    99.34/44.89
POL(c(x1, x2)) = x1 + x2    99.34/44.89
POL(c3(x1)) = x1    99.34/44.89
POL(false) = [3]    99.34/44.89
POL(gr(x1, x2)) = 0    99.34/44.89
POL(s(x1)) = [1] + x1    99.34/44.89
POL(true) = 0   
99.34/44.89
99.34/44.89

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), z1) 99.34/44.89
gr(0, z0) → false 99.34/44.89
gr(s(z0), 0) → true 99.34/44.89
gr(s(z0), s(z1)) → gr(z0, z1) 99.34/44.89
p(0) → 0 99.34/44.89
p(s(z0)) → z0
Tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0))) 99.34/44.89
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))))
S tuples:none
K tuples:

COND(true, 0, z0) → c 99.34/44.89
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(z1)), GR(s(z0), s(z1))) 99.34/44.89
COND(true, s(z0), 0) → c(COND(true, z0, 0)) 99.34/44.89
COND(true, s(0), s(z0)) → c 99.34/44.89
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(0))) 99.34/44.89
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)))) 99.34/44.89
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c, c3

99.34/44.89
99.34/44.89

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

The set S is empty
99.34/44.89
99.34/44.89

(64) BOUNDS(O(1), O(1))

99.34/44.89
99.34/44.89
99.69/44.92 EOF