YES(O(1), O(n^1)) 0.00/0.88 YES(O(1), O(n^1)) 0.00/0.91 0.00/0.91 0.00/0.91 0.00/0.91 0.00/0.91 0.00/0.91 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.91 0.00/0.91 0.00/0.91
0.00/0.91 0.00/0.91 0.00/0.91
0.00/0.91
0.00/0.91

(0) Obligation:

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

monus(S(x'), S(x)) → monus(x', x) 0.00/0.91
gcd(x, y) → gcd[Ite][False][Ite][False][Ite](equal0(x, y), x, y) 0.00/0.91
equal0(a, b) → equal0[Ite](<(a, b), a, b)

The (relative) TRS S consists of the following rules:

<(S(x), S(y)) → <(x, y) 0.00/0.91
<(0, S(y)) → True 0.00/0.91
<(x, 0) → False 0.00/0.91
equal0[Ite](False, a, b) → False 0.00/0.91
equal0[Ite](True, a, b) → equal0[Ite][True][Ite](<(b, a), a, b) 0.00/0.91
equal0[Ite][True][Ite](False, a, b) → False 0.00/0.91
equal0[Ite][True][Ite](True, a, b) → True

Rewrite Strategy: INNERMOST
0.00/0.91
0.00/0.91

(1) CpxRelTrsToCDT (UPPER BOUND (ID) transformation)

Relative innermost TRS to CDT Problem.
0.00/0.91
0.00/0.91

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

<(S(z0), S(z1)) → <(z0, z1) 0.00/0.91
<(0, S(z0)) → True 0.00/0.91
<(z0, 0) → False 0.00/0.91
equal0[Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite](True, z0, z1) → equal0[Ite][True][Ite](<(z1, z0), z0, z1) 0.00/0.91
equal0[Ite][True][Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite][True][Ite](True, z0, z1) → True 0.00/0.91
monus(S(z0), S(z1)) → monus(z0, z1) 0.00/0.91
gcd(z0, z1) → gcd[Ite][False][Ite][False][Ite](equal0(z0, z1), z0, z1) 0.00/0.91
equal0(z0, z1) → equal0[Ite](<(z0, z1), z0, z1)
Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(EQUAL0[ITE][TRUE][ITE](<(z1, z0), z0, z1), <'(z1, z0)) 0.00/0.91
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
GCD(z0, z1) → c8(EQUAL0(z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c9(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
S tuples:

MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
GCD(z0, z1) → c8(EQUAL0(z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c9(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
K tuples:none
Defined Rule Symbols:

monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]

Defined Pair Symbols:

<', EQUAL0[ITE], MONUS, GCD, EQUAL0

Compound Symbols:

c, c4, c7, c8, c9

0.00/0.91
0.00/0.91

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

Removed 1 trailing tuple parts
0.00/0.91
0.00/0.91

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

<(S(z0), S(z1)) → <(z0, z1) 0.00/0.91
<(0, S(z0)) → True 0.00/0.91
<(z0, 0) → False 0.00/0.91
equal0[Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite](True, z0, z1) → equal0[Ite][True][Ite](<(z1, z0), z0, z1) 0.00/0.91
equal0[Ite][True][Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite][True][Ite](True, z0, z1) → True 0.00/0.91
monus(S(z0), S(z1)) → monus(z0, z1) 0.00/0.91
gcd(z0, z1) → gcd[Ite][False][Ite][False][Ite](equal0(z0, z1), z0, z1) 0.00/0.91
equal0(z0, z1) → equal0[Ite](<(z0, z1), z0, z1)
Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 0.00/0.91
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
GCD(z0, z1) → c8(EQUAL0(z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c9(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0))
S tuples:

MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
GCD(z0, z1) → c8(EQUAL0(z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c9(EQUAL0[ITE](<(z0, z1), z0, z1), <'(z0, z1))
K tuples:none
Defined Rule Symbols:

monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]

Defined Pair Symbols:

<', MONUS, GCD, EQUAL0, EQUAL0[ITE]

Compound Symbols:

c, c7, c8, c9, c4

0.00/0.91
0.00/0.91

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

Split RHS of tuples not part of any SCC
0.00/0.91
0.00/0.91

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

<(S(z0), S(z1)) → <(z0, z1) 0.00/0.91
<(0, S(z0)) → True 0.00/0.91
<(z0, 0) → False 0.00/0.91
equal0[Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite](True, z0, z1) → equal0[Ite][True][Ite](<(z1, z0), z0, z1) 0.00/0.91
equal0[Ite][True][Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite][True][Ite](True, z0, z1) → True 0.00/0.91
monus(S(z0), S(z1)) → monus(z0, z1) 0.00/0.91
gcd(z0, z1) → gcd[Ite][False][Ite][False][Ite](equal0(z0, z1), z0, z1) 0.00/0.91
equal0(z0, z1) → equal0[Ite](<(z0, z1), z0, z1)
Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 0.00/0.91
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
GCD(z0, z1) → c8(EQUAL0(z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0)) 0.00/0.91
EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c1(<'(z0, z1))
S tuples:

MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
GCD(z0, z1) → c8(EQUAL0(z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c1(<'(z0, z1))
K tuples:none
Defined Rule Symbols:

monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]

Defined Pair Symbols:

<', MONUS, GCD, EQUAL0[ITE], EQUAL0

Compound Symbols:

c, c7, c8, c4, c1

0.00/0.91
0.00/0.91

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

GCD(z0, z1) → c8(EQUAL0(z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c1(<'(z0, z1))
0.00/0.91
0.00/0.91

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

<(S(z0), S(z1)) → <(z0, z1) 0.00/0.91
<(0, S(z0)) → True 0.00/0.91
<(z0, 0) → False 0.00/0.91
equal0[Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite](True, z0, z1) → equal0[Ite][True][Ite](<(z1, z0), z0, z1) 0.00/0.91
equal0[Ite][True][Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite][True][Ite](True, z0, z1) → True 0.00/0.91
monus(S(z0), S(z1)) → monus(z0, z1) 0.00/0.91
gcd(z0, z1) → gcd[Ite][False][Ite][False][Ite](equal0(z0, z1), z0, z1) 0.00/0.91
equal0(z0, z1) → equal0[Ite](<(z0, z1), z0, z1)
Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 0.00/0.91
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0)) 0.00/0.91
EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1))
S tuples:

MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1))
K tuples:none
Defined Rule Symbols:

monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]

Defined Pair Symbols:

<', MONUS, EQUAL0[ITE], EQUAL0

Compound Symbols:

c, c7, c4, c1

0.00/0.91
0.00/0.91

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

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

EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0))
0.00/0.91
0.00/0.91

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

<(S(z0), S(z1)) → <(z0, z1) 0.00/0.91
<(0, S(z0)) → True 0.00/0.91
<(z0, 0) → False 0.00/0.91
equal0[Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite](True, z0, z1) → equal0[Ite][True][Ite](<(z1, z0), z0, z1) 0.00/0.91
equal0[Ite][True][Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite][True][Ite](True, z0, z1) → True 0.00/0.91
monus(S(z0), S(z1)) → monus(z0, z1) 0.00/0.91
gcd(z0, z1) → gcd[Ite][False][Ite][False][Ite](equal0(z0, z1), z0, z1) 0.00/0.91
equal0(z0, z1) → equal0[Ite](<(z0, z1), z0, z1)
Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 0.00/0.91
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0)) 0.00/0.91
EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1))
S tuples:

MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1))
K tuples:

EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0))
Defined Rule Symbols:

monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]

Defined Pair Symbols:

<', MONUS, EQUAL0[ITE], EQUAL0

Compound Symbols:

c, c7, c4, c1

0.00/0.91
0.00/0.91

(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.

MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1))
We considered the (Usable) Rules:

<(S(z0), S(z1)) → <(z0, z1) 0.00/0.91
<(0, S(z0)) → True 0.00/0.91
<(z0, 0) → False
And the Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 0.00/0.91
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0)) 0.00/0.91
EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.91

POL(0) = [2]    0.00/0.91
POL(<(x1, x2)) = [2] + [2]x1 + [4]x2    0.00/0.91
POL(<'(x1, x2)) = [4] + x1 + x2    0.00/0.91
POL(EQUAL0(x1, x2)) = [5] + [5]x1 + x2    0.00/0.91
POL(EQUAL0[ITE](x1, x2, x3)) = [5] + [5]x2 + x3    0.00/0.91
POL(False) = [5]    0.00/0.91
POL(MONUS(x1, x2)) = x2    0.00/0.91
POL(S(x1)) = [1] + x1    0.00/0.91
POL(True) = [5]    0.00/0.91
POL(c(x1)) = x1    0.00/0.91
POL(c1(x1)) = x1    0.00/0.91
POL(c4(x1)) = x1    0.00/0.91
POL(c7(x1)) = x1   
0.00/0.91
0.00/0.91

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

<(S(z0), S(z1)) → <(z0, z1) 0.00/0.91
<(0, S(z0)) → True 0.00/0.91
<(z0, 0) → False 0.00/0.91
equal0[Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite](True, z0, z1) → equal0[Ite][True][Ite](<(z1, z0), z0, z1) 0.00/0.91
equal0[Ite][True][Ite](False, z0, z1) → False 0.00/0.91
equal0[Ite][True][Ite](True, z0, z1) → True 0.00/0.91
monus(S(z0), S(z1)) → monus(z0, z1) 0.00/0.91
gcd(z0, z1) → gcd[Ite][False][Ite][False][Ite](equal0(z0, z1), z0, z1) 0.00/0.91
equal0(z0, z1) → equal0[Ite](<(z0, z1), z0, z1)
Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 0.00/0.91
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0)) 0.00/0.91
EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1))
S tuples:none
K tuples:

EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0)) 0.00/0.91
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1))
Defined Rule Symbols:

monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]

Defined Pair Symbols:

<', MONUS, EQUAL0[ITE], EQUAL0

Compound Symbols:

c, c7, c4, c1

0.00/0.91
0.00/0.91

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

The set S is empty
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0.00/0.91

(14) BOUNDS(O(1), O(1))

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0.00/0.91
0.00/0.95 EOF