01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
BS1(n3(x, y, z)) -> GE2(x, max1(y))
-12(01(x), 11(y)) -> -12(-2(x, y), 11(#))
GE2(11(x), 01(y)) -> GE2(x, y)
GE2(01(x), 11(y)) -> GE2(y, x)
SIZE1(n3(x, y, z)) -> +12(+2(size1(x), size1(y)), 11(#))
+12(x, +2(y, z)) -> +12(+2(x, y), z)
+12(11(x), 11(y)) -> 011(+2(+2(x, y), 11(#)))
-12(11(x), 01(y)) -> -12(x, y)
-12(01(x), 11(y)) -> -12(x, y)
SIZE1(n3(x, y, z)) -> +12(size1(x), size1(y))
BS1(n3(x, y, z)) -> BS1(z)
WB1(n3(x, y, z)) -> GE2(11(#), -2(size1(y), size1(z)))
WB1(n3(x, y, z)) -> GE2(11(#), -2(size1(z), size1(y)))
WB1(n3(x, y, z)) -> AND2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
BS1(n3(x, y, z)) -> AND2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
GE2(01(x), 11(y)) -> NOT1(ge2(y, x))
SIZE1(n3(x, y, z)) -> SIZE1(y)
+12(x, +2(y, z)) -> +12(x, y)
SIZE1(n3(x, y, z)) -> SIZE1(x)
WB1(n3(x, y, z)) -> SIZE1(y)
WB1(n3(x, y, z)) -> SIZE1(z)
BS1(n3(x, y, z)) -> BS1(y)
MIN1(n3(x, y, z)) -> MIN1(y)
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 01(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
WB1(n3(x, y, z)) -> AND2(wb1(y), wb1(z))
-12(11(x), 11(y)) -> 011(-2(x, y))
BS1(n3(x, y, z)) -> MAX1(y)
WB1(n3(x, y, z)) -> GE2(size1(y), size1(z))
+12(01(x), 01(y)) -> +12(x, y)
-12(01(x), 01(y)) -> 011(-2(x, y))
MAX1(n3(x, y, z)) -> MAX1(z)
WB1(n3(x, y, z)) -> -12(size1(y), size1(z))
WB1(n3(x, y, z)) -> -12(size1(z), size1(y))
BS1(n3(x, y, z)) -> AND2(ge2(x, max1(y)), ge2(min1(z), x))
+12(01(x), 01(y)) -> 011(+2(x, y))
WB1(n3(x, y, z)) -> WB1(z)
GE2(11(x), 11(y)) -> GE2(x, y)
+12(11(x), 11(y)) -> +12(x, y)
-12(01(x), 01(y)) -> -12(x, y)
WB1(n3(x, y, z)) -> WB1(y)
GE2(#, 01(x)) -> GE2(#, x)
BS1(n3(x, y, z)) -> MIN1(z)
BS1(n3(x, y, z)) -> GE2(min1(z), x)
BS1(n3(x, y, z)) -> AND2(bs1(y), bs1(z))
WB1(n3(x, y, z)) -> IF3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y))))
GE2(01(x), 01(y)) -> GE2(x, y)
-12(11(x), 11(y)) -> -12(x, y)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
BS1(n3(x, y, z)) -> GE2(x, max1(y))
-12(01(x), 11(y)) -> -12(-2(x, y), 11(#))
GE2(11(x), 01(y)) -> GE2(x, y)
GE2(01(x), 11(y)) -> GE2(y, x)
SIZE1(n3(x, y, z)) -> +12(+2(size1(x), size1(y)), 11(#))
+12(x, +2(y, z)) -> +12(+2(x, y), z)
+12(11(x), 11(y)) -> 011(+2(+2(x, y), 11(#)))
-12(11(x), 01(y)) -> -12(x, y)
-12(01(x), 11(y)) -> -12(x, y)
SIZE1(n3(x, y, z)) -> +12(size1(x), size1(y))
BS1(n3(x, y, z)) -> BS1(z)
WB1(n3(x, y, z)) -> GE2(11(#), -2(size1(y), size1(z)))
WB1(n3(x, y, z)) -> GE2(11(#), -2(size1(z), size1(y)))
WB1(n3(x, y, z)) -> AND2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
BS1(n3(x, y, z)) -> AND2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
GE2(01(x), 11(y)) -> NOT1(ge2(y, x))
SIZE1(n3(x, y, z)) -> SIZE1(y)
+12(x, +2(y, z)) -> +12(x, y)
SIZE1(n3(x, y, z)) -> SIZE1(x)
WB1(n3(x, y, z)) -> SIZE1(y)
WB1(n3(x, y, z)) -> SIZE1(z)
BS1(n3(x, y, z)) -> BS1(y)
MIN1(n3(x, y, z)) -> MIN1(y)
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 01(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
WB1(n3(x, y, z)) -> AND2(wb1(y), wb1(z))
-12(11(x), 11(y)) -> 011(-2(x, y))
BS1(n3(x, y, z)) -> MAX1(y)
WB1(n3(x, y, z)) -> GE2(size1(y), size1(z))
+12(01(x), 01(y)) -> +12(x, y)
-12(01(x), 01(y)) -> 011(-2(x, y))
MAX1(n3(x, y, z)) -> MAX1(z)
WB1(n3(x, y, z)) -> -12(size1(y), size1(z))
WB1(n3(x, y, z)) -> -12(size1(z), size1(y))
BS1(n3(x, y, z)) -> AND2(ge2(x, max1(y)), ge2(min1(z), x))
+12(01(x), 01(y)) -> 011(+2(x, y))
WB1(n3(x, y, z)) -> WB1(z)
GE2(11(x), 11(y)) -> GE2(x, y)
+12(11(x), 11(y)) -> +12(x, y)
-12(01(x), 01(y)) -> -12(x, y)
WB1(n3(x, y, z)) -> WB1(y)
GE2(#, 01(x)) -> GE2(#, x)
BS1(n3(x, y, z)) -> MIN1(z)
BS1(n3(x, y, z)) -> GE2(min1(z), x)
BS1(n3(x, y, z)) -> AND2(bs1(y), bs1(z))
WB1(n3(x, y, z)) -> IF3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y))))
GE2(01(x), 01(y)) -> GE2(x, y)
-12(11(x), 11(y)) -> -12(x, y)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
MAX1(n3(x, y, z)) -> MAX1(z)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MAX1(n3(x, y, z)) -> MAX1(z)
POL(MAX1(x1)) = x1
POL(n3(x1, x2, x3)) = 1 + x3
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
MIN1(n3(x, y, z)) -> MIN1(y)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MIN1(n3(x, y, z)) -> MIN1(y)
POL(MIN1(x1)) = x1
POL(n3(x1, x2, x3)) = 1 + x2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
GE2(#, 01(x)) -> GE2(#, x)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
GE2(#, 01(x)) -> GE2(#, x)
POL(#) = 0
POL(01(x1)) = 1 + x1
POL(GE2(x1, x2)) = x2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
GE2(11(x), 11(y)) -> GE2(x, y)
GE2(01(x), 01(y)) -> GE2(x, y)
GE2(11(x), 01(y)) -> GE2(x, y)
GE2(01(x), 11(y)) -> GE2(y, x)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
GE2(11(x), 11(y)) -> GE2(x, y)
GE2(11(x), 01(y)) -> GE2(x, y)
GE2(01(x), 11(y)) -> GE2(y, x)
Used ordering: Polynomial interpretation [21]:
GE2(01(x), 01(y)) -> GE2(x, y)
POL(01(x1)) = x1
POL(11(x1)) = 1 + x1
POL(GE2(x1, x2)) = x1 + x2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
GE2(01(x), 01(y)) -> GE2(x, y)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
GE2(01(x), 01(y)) -> GE2(x, y)
POL(01(x1)) = 1 + x1
POL(GE2(x1, x2)) = x2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
BS1(n3(x, y, z)) -> BS1(y)
BS1(n3(x, y, z)) -> BS1(z)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
BS1(n3(x, y, z)) -> BS1(y)
BS1(n3(x, y, z)) -> BS1(z)
POL(BS1(x1)) = x1
POL(n3(x1, x2, x3)) = 1 + x2 + x3
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
-12(11(x), 01(y)) -> -12(x, y)
-12(01(x), 11(y)) -> -12(x, y)
-12(01(x), 01(y)) -> -12(x, y)
-12(01(x), 11(y)) -> -12(-2(x, y), 11(#))
-12(11(x), 11(y)) -> -12(x, y)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
-12(11(x), 01(y)) -> -12(x, y)
-12(01(x), 01(y)) -> -12(x, y)
Used ordering: Polynomial interpretation [21]:
-12(01(x), 11(y)) -> -12(x, y)
-12(01(x), 11(y)) -> -12(-2(x, y), 11(#))
-12(11(x), 11(y)) -> -12(x, y)
POL(#) = 0
POL(-2(x1, x2)) = 0
POL(-12(x1, x2)) = x2
POL(01(x1)) = 1 + x1
POL(11(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
-12(01(x), 11(y)) -> -12(x, y)
-12(01(x), 11(y)) -> -12(-2(x, y), 11(#))
-12(11(x), 11(y)) -> -12(x, y)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
-12(01(x), 11(y)) -> -12(x, y)
-12(11(x), 11(y)) -> -12(x, y)
Used ordering: Polynomial interpretation [21]:
-12(01(x), 11(y)) -> -12(-2(x, y), 11(#))
POL(#) = 0
POL(-2(x1, x2)) = 0
POL(-12(x1, x2)) = x2
POL(01(x1)) = 0
POL(11(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
-12(01(x), 11(y)) -> -12(-2(x, y), 11(#))
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
-12(01(x), 11(y)) -> -12(-2(x, y), 11(#))
POL(#) = 0
POL(-2(x1, x2)) = x1
POL(-12(x1, x2)) = x1
POL(01(x1)) = 1 + x1
POL(11(x1)) = 1 + x1
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(x, #) -> x
01(#) -> #
-2(11(x), 11(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(#, x) -> #
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
+12(01(x), 01(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(x, y)
+12(x, +2(y, z)) -> +12(x, y)
+12(11(x), 01(y)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(x, +2(y, z)) -> +12(+2(x, y), z)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+12(01(x), 01(y)) -> +12(x, y)
+12(11(x), 01(y)) -> +12(x, y)
Used ordering: Polynomial interpretation [21]:
+12(11(x), 11(y)) -> +12(x, y)
+12(x, +2(y, z)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(x, +2(y, z)) -> +12(+2(x, y), z)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
POL(#) = 0
POL(+2(x1, x2)) = x1 + x2
POL(+12(x1, x2)) = x2
POL(01(x1)) = 1 + x1
POL(11(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
+12(11(x), 11(y)) -> +12(x, y)
+12(x, +2(y, z)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(x, +2(y, z)) -> +12(+2(x, y), z)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+12(x, +2(y, z)) -> +12(x, y)
+12(x, +2(y, z)) -> +12(+2(x, y), z)
Used ordering: Polynomial interpretation [21]:
+12(11(x), 11(y)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
POL(#) = 0
POL(+2(x1, x2)) = 1 + x1 + x2
POL(+12(x1, x2)) = x2
POL(01(x1)) = 0
POL(11(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
+12(11(x), 11(y)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+12(11(x), 11(y)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
Used ordering: Polynomial interpretation [21]:
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
POL(#) = 0
POL(+2(x1, x2)) = 0
POL(+12(x1, x2)) = x2
POL(01(x1)) = 0
POL(11(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
POL(#) = 0
POL(+2(x1, x2)) = x1 + x2
POL(+12(x1, x2)) = x1 + x2
POL(01(x1)) = x1
POL(11(x1)) = 1 + x1
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
+2(11(x), 01(y)) -> 11(+2(x, y))
01(#) -> #
+2(#, x) -> x
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, #) -> x
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
SIZE1(n3(x, y, z)) -> SIZE1(y)
SIZE1(n3(x, y, z)) -> SIZE1(x)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SIZE1(n3(x, y, z)) -> SIZE1(y)
SIZE1(n3(x, y, z)) -> SIZE1(x)
POL(SIZE1(x1)) = x1
POL(n3(x1, x2, x3)) = 1 + x1 + x2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
WB1(n3(x, y, z)) -> WB1(y)
WB1(n3(x, y, z)) -> WB1(z)
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
WB1(n3(x, y, z)) -> WB1(y)
WB1(n3(x, y, z)) -> WB1(z)
POL(WB1(x1)) = x1
POL(n3(x1, x2, x3)) = 1 + x2 + x3
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(x, +2(y, z)) -> +2(+2(x, y), z)
-2(x, #) -> x
-2(#, x) -> #
-2(01(x), 01(y)) -> 01(-2(x, y))
-2(01(x), 11(y)) -> 11(-2(-2(x, y), 11(#)))
-2(11(x), 01(y)) -> 11(-2(x, y))
-2(11(x), 11(y)) -> 01(-2(x, y))
not1(false) -> true
not1(true) -> false
and2(x, true) -> x
and2(x, false) -> false
if3(true, x, y) -> x
if3(false, x, y) -> y
ge2(01(x), 01(y)) -> ge2(x, y)
ge2(01(x), 11(y)) -> not1(ge2(y, x))
ge2(11(x), 01(y)) -> ge2(x, y)
ge2(11(x), 11(y)) -> ge2(x, y)
ge2(x, #) -> true
ge2(#, 11(x)) -> false
ge2(#, 01(x)) -> ge2(#, x)
val1(l1(x)) -> x
val1(n3(x, y, z)) -> x
min1(l1(x)) -> x
min1(n3(x, y, z)) -> min1(y)
max1(l1(x)) -> x
max1(n3(x, y, z)) -> max1(z)
bs1(l1(x)) -> true
bs1(n3(x, y, z)) -> and2(and2(ge2(x, max1(y)), ge2(min1(z), x)), and2(bs1(y), bs1(z)))
size1(l1(x)) -> 11(#)
size1(n3(x, y, z)) -> +2(+2(size1(x), size1(y)), 11(#))
wb1(l1(x)) -> true
wb1(n3(x, y, z)) -> and2(if3(ge2(size1(y), size1(z)), ge2(11(#), -2(size1(y), size1(z))), ge2(11(#), -2(size1(z), size1(y)))), and2(wb1(y), wb1(z)))