minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
↳ QTRS
↳ DependencyPairsProof
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
GCD(s(X), s(Y)) → IF(le(Y, X), s(X), s(Y))
IF(true, s(X), s(Y)) → MINUS(X, Y)
IF(true, s(X), s(Y)) → GCD(minus(X, Y), s(Y))
IF(false, s(X), s(Y)) → GCD(minus(Y, X), s(X))
MINUS(X, s(Y)) → MINUS(X, Y)
IF(false, s(X), s(Y)) → MINUS(Y, X)
LE(s(X), s(Y)) → LE(X, Y)
GCD(s(X), s(Y)) → LE(Y, X)
MINUS(X, s(Y)) → PRED(minus(X, Y))
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
GCD(s(X), s(Y)) → IF(le(Y, X), s(X), s(Y))
IF(true, s(X), s(Y)) → MINUS(X, Y)
IF(true, s(X), s(Y)) → GCD(minus(X, Y), s(Y))
IF(false, s(X), s(Y)) → GCD(minus(Y, X), s(X))
MINUS(X, s(Y)) → MINUS(X, Y)
IF(false, s(X), s(Y)) → MINUS(Y, X)
LE(s(X), s(Y)) → LE(X, Y)
GCD(s(X), s(Y)) → LE(Y, X)
MINUS(X, s(Y)) → PRED(minus(X, Y))
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
LE(s(X), s(Y)) → LE(X, Y)
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
LE(s(X), s(Y)) → LE(X, Y)
The value of delta used in the strict ordering is 3.
POL(s(x1)) = 1 + (4)x_1
POL(LE(x1, x2)) = (3)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
MINUS(X, s(Y)) → MINUS(X, Y)
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MINUS(X, s(Y)) → MINUS(X, Y)
The value of delta used in the strict ordering is 4.
POL(MINUS(x1, x2)) = (4)x_2
POL(s(x1)) = 1 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
GCD(s(X), s(Y)) → IF(le(Y, X), s(X), s(Y))
IF(true, s(X), s(Y)) → GCD(minus(X, Y), s(Y))
IF(false, s(X), s(Y)) → GCD(minus(Y, X), s(X))
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
IF(false, s(X), s(Y)) → GCD(minus(Y, X), s(X))
Used ordering: Polynomial interpretation [25,35]:
GCD(s(X), s(Y)) → IF(le(Y, X), s(X), s(Y))
IF(true, s(X), s(Y)) → GCD(minus(X, Y), s(Y))
The value of delta used in the strict ordering is 4.
POL(minus(x1, x2)) = (2)x_1
POL(le(x1, x2)) = (2)x_2
POL(true) = 0
POL(pred(x1)) = x_1
POL(false) = 1
POL(s(x1)) = (4)x_1
POL(GCD(x1, x2)) = (4)x_1 + (2)x_2
POL(IF(x1, x2, x3)) = (4)x_1 + (2)x_2 + (2)x_3
POL(0) = 1
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
GCD(s(X), s(Y)) → IF(le(Y, X), s(X), s(Y))
IF(true, s(X), s(Y)) → GCD(minus(X, Y), s(Y))
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
GCD(s(X), s(Y)) → IF(le(Y, X), s(X), s(Y))
Used ordering: Polynomial interpretation [25,35]:
IF(true, s(X), s(Y)) → GCD(minus(X, Y), s(Y))
The value of delta used in the strict ordering is 1.
POL(minus(x1, x2)) = x_1
POL(le(x1, x2)) = (2)x_2
POL(true) = 0
POL(pred(x1)) = x_1
POL(false) = 2
POL(s(x1)) = 1 + (2)x_1
POL(GCD(x1, x2)) = 1 + x_1
POL(IF(x1, x2, x3)) = x_2
POL(0) = 0
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
IF(true, s(X), s(Y)) → GCD(minus(X, Y), s(Y))
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))