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 0.00/0.910 CpxRelTRS0.00/0.91
↳1 CpxRelTrsToCDT (UPPER BOUND (ID))0.00/0.91
↳2 CdtProblem0.00/0.91
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))0.00/0.91
↳4 CdtProblem0.00/0.91
↳5 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))0.00/0.91
↳6 CdtProblem0.00/0.91
↳7 CdtLeafRemovalProof (ComplexityIfPolyImplication)0.00/0.91
↳8 CdtProblem0.00/0.91
↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))0.00/0.91
↳10 CdtProblem0.00/0.91
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))0.00/0.91
↳12 CdtProblem0.00/0.91
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))0.00/0.91
↳14 BOUNDS(O(1), O(1))0.00/0.91
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)
<(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
Tuples:
<(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)
S 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))
K tuples:none
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))
monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]
<', EQUAL0[ITE], MONUS, GCD, EQUAL0
c, c4, c7, c8, c9
Tuples:
<(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)
S 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))
K tuples:none
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))
monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]
<', MONUS, GCD, EQUAL0, EQUAL0[ITE]
c, c7, c8, c9, c4
Tuples:
<(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)
S 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))
K tuples:none
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))
monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]
<', MONUS, GCD, EQUAL0[ITE], EQUAL0
c, c7, c8, c4, c1
GCD(z0, z1) → c8(EQUAL0(z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c1(<'(z0, z1))
Tuples:
<(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)
S 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))
K tuples:none
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1)) 0.00/0.91
EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1))
monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]
<', MONUS, EQUAL0[ITE], EQUAL0
c, c7, c4, c1
EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0))
Tuples:
<(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)
S 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))
K tuples:
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1))
Defined Rule Symbols:
EQUAL0(z0, z1) → c1(EQUAL0[ITE](<(z0, z1), z0, z1)) 0.00/0.91
EQUAL0[ITE](True, z0, z1) → c4(<'(z1, z0))
monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]
<', MONUS, EQUAL0[ITE], EQUAL0
c, c7, c4, c1
We considered the (Usable) Rules:
MONUS(S(z0), S(z1)) → c7(MONUS(z0, z1))
And the Tuples:
<(S(z0), S(z1)) → <(z0, z1) 0.00/0.91
<(0, S(z0)) → True 0.00/0.91
<(z0, 0) → False
The order we found is given by the following interpretation:
<'(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))
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
Tuples:
<(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)
S tuples:none
<'(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))
Defined Rule Symbols:
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))
monus, gcd, equal0, <, equal0[Ite], equal0[Ite][True][Ite]
<', MONUS, EQUAL0[ITE], EQUAL0
c, c7, c4, c1