le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
↳ QTRS
↳ DependencyPairsProof
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
QUOT(s(X), s(Y)) → QUOT(minus(X, Y), s(Y))
IFMINUS(false, s(X), Y) → MINUS(X, Y)
MINUS(s(X), Y) → LE(s(X), Y)
MINUS(s(X), Y) → IFMINUS(le(s(X), Y), s(X), Y)
LE(s(X), s(Y)) → LE(X, Y)
QUOT(s(X), s(Y)) → MINUS(X, Y)
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
QUOT(s(X), s(Y)) → QUOT(minus(X, Y), s(Y))
IFMINUS(false, s(X), Y) → MINUS(X, Y)
MINUS(s(X), Y) → LE(s(X), Y)
MINUS(s(X), Y) → IFMINUS(le(s(X), Y), s(X), Y)
LE(s(X), s(Y)) → LE(X, Y)
QUOT(s(X), s(Y)) → MINUS(X, Y)
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
LE(s(X), s(Y)) → LE(X, Y)
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
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 15/8.
POL(s(x1)) = 1/2 + (13/4)x_1
POL(LE(x1, x2)) = (15/4)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
IFMINUS(false, s(X), Y) → MINUS(X, Y)
MINUS(s(X), Y) → IFMINUS(le(s(X), Y), s(X), Y)
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
IFMINUS(false, s(X), Y) → MINUS(X, Y)
Used ordering: Polynomial interpretation [25,35]:
MINUS(s(X), Y) → IFMINUS(le(s(X), Y), s(X), Y)
The value of delta used in the strict ordering is 3/16.
POL(IFMINUS(x1, x2, x3)) = (4)x_1 + (3/4)x_2 + (1/2)x_3
POL(le(x1, x2)) = 1/4
POL(true) = 1/4
POL(false) = 1/4
POL(MINUS(x1, x2)) = 1 + (3/4)x_1 + (1/2)x_2
POL(s(x1)) = 1/4 + (9/4)x_1
POL(0) = 13/4
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
MINUS(s(X), Y) → IFMINUS(le(s(X), Y), s(X), Y)
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
QUOT(s(X), s(Y)) → QUOT(minus(X, Y), s(Y))
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
QUOT(s(X), s(Y)) → QUOT(minus(X, Y), s(Y))
The value of delta used in the strict ordering is 3/4.
POL(minus(x1, x2)) = 1/2 + (2)x_1
POL(ifMinus(x1, x2, x3)) = 1/2 + (2)x_2
POL(QUOT(x1, x2)) = (1/2)x_1
POL(le(x1, x2)) = 1 + (2)x_1
POL(true) = 15/4
POL(false) = 5/2
POL(s(x1)) = 2 + (3)x_1
POL(0) = 15/4
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))