minRealPosClosure
maxRealClosure = Forall((a, b), InSet(Max(a, b), Reals), domain=Reals)
maxRealClosure
maxRealPosClosure = Forall((a, b), InSet(Max(a, b), RealsPos), domain=RealsPos)
maxRealPosClosure
relaxGreaterThan = Forall([a, b],
GreaterThanEquals(a, b),
domain=Reals,
conditions=GreaterThan(a, b))
relaxGreaterThan
relaxLessThan = Forall([a, b],
LessThanEquals(a, b),
domain=Reals,
conditions=LessThan(a, b))
relaxLessThan
lessThanInBools = Forall([a, b], InSet(LessThan(a, b), Booleans), domain=Reals)
lessThanInBools
lessThanEqualsInBools = Forall([a, b],
InSet(LessThanEquals(a, b), Booleans),
domain=Reals)
lessThanEqualsInBools
greaterThanInBools = Forall([a, b],
InSet(GreaterThan(a, b), Booleans),
domain=Reals)
allInIntervalCO_InReals = Forall((a, b), Forall(x, InSet(x, Reals), domain=IntervalCO(a, b)), domain=Reals) allInIntervalCO_InReals allInIntervalOC_InReals = Forall((a, b), Forall(x, InSet(x, Reals), domain=IntervalOC(a, b)), domain=Reals) allInIntervalOC_InReals allInIntervalCC_InReals = Forall((a, b), Forall(x, InSet(x, Reals), domain=IntervalCC(a, b)), domain=Reals) allInIntervalCC_InReals intervalOOLowerBound = Forall((a, b), Forall(x, LessThan(a, x), domain=IntervalOO(a, b)), domain=Reals) intervalOOLowerBound intervalOOUpperBound = Forall((a, b), Forall(x, LessThan(x, b), domain=IntervalOO(a, b)), domain=Reals) intervalOOUpperBound intervalCOLowerBound = Forall((a, b), Forall(x, LessThanEquals(a, x), domain=IntervalCO(a, b)), domain=Reals) intervalCOLowerBound intervalCOUpperBound = Forall((a, b), Forall(x, LessThan(x, b), domain=IntervalCO(a, b)), domain=Reals) intervalCOUpperBound intervalOCLowerBound = Forall((a, b), Forall(x, LessThan(a, x), domain=IntervalOC(a, b)), domain=Reals) intervalOCLowerBound intervalOCUpperBound = Forall((a, b), Forall(x, LessThanEquals(x, b), domain=IntervalOC(a, b)), domain=Reals) intervalOCUpperBound intervalCCLowerBound = Forall((a, b), Forall(x, LessThanEquals(a, x), domain=IntervalCC(a, b)), domain=Reals) intervalCCLowerBound intervalCCUpperBound = Forall((a, b), Forall(x, LessThanEquals(x, b), domain=IntervalCC(a, b)), domain=Reals)
modInIntervalCO
absIsNonNeg = Forall(a, GreaterThanEquals(Abs(a), zero), domain=Complexes)
absIsNonNeg
absNotEqZero = Forall([a],
NotEquals(Abs(a), zero),
domain=Complexes,
conditions=[NotEquals(a, zero)])
absNotEqZero
absElim = Forall(x, Equals(Abs(x), x), domain=RealsPos)
absElim
absIneq = Forall((x, y),
Iff(LessThanEquals(Abs(x), y),
And(LessThanEquals(Neg(y), x), LessThanEquals(x, y))),
domain=Reals,
conditions=[GreaterThanEquals(y, zero)])
absIneq
triangleInequality = Forall([a, b],
LessThanEquals(Abs(Add(a, b)), Add(Abs(a),
Abs(b))),
domain=Complexes)
triangleInequality
absProd = Forall(xEtc,
Equals(Abs(Mult(xEtc)), Mult(Etcetera(Abs(xMulti)))),
domain=Complexes)
absProd
sqrtRealPosClosure = Forall([a], InSet(sqrt(a), RealsPos), domain=RealsPos)
sqrtRealPosClosure
sqrtComplexClosure = Forall([a], InSet(sqrt(a), Complexes), domain=Complexes)
sqrtComplexClosure
# Should generalize to even power closure, but need to define and implement evens set to do this.
sqrdPosClosure = Forall(a,
InSet(Exp(a, two), RealsPos),
domain=Reals,
conditions=[NotEquals(a, zero)])
sqrdPosClosure
squarePosIneq = Forall([a, b],
LessThanEquals(Exp(Abs(a), two), Exp(b, two)),
domain=Reals,
conditions=(LessThanEquals(Abs(a), b), ))
squarePosIneq
squarePosEq = Forall(a, Equals(Exp(Abs(a), two), Exp(a, two)), domain=Reals)
squarePosEq
expNotEqZero = Forall([a, b],
NotEquals(Exp(a, b), zero),
domain=Complexes,
conditions=[NotEquals(a, zero)])
expNotEqZero
expZeroEqOne = Forall([a],
Equals(Exp(a, zero), one),
Forall(xMulti,
InSet(PxEtc, Complexes),
domain=S),
InSet(Sum(xMulti, PxEtc, domain=S),
Complexes)))
summationComplexClosure
sumSplitAfter = Forall(
f,
Forall([a, b, c],
Equals(
Sum(x, fx, Interval(a, c)),
Add(Sum(x, fx, Interval(a, b)),
Sum(x, fx, Interval(Add(b, one), c)))),
domain=Integers,
conditions=[LessThanEquals(a, b),
LessThan(b, c)]))
sumSplitAfter
sumSplitBefore = Forall(
f,
Forall([a, b, c],
Equals(
Sum(x, fx, Interval(a, c)),
Add(Sum(x, fx, Interval(a, Sub(b, one))),
Sum(x, fx, Interval(b, c)))),
domain=Integers,
conditions=[LessThan(a, b), LessThanEquals(b, c)]))
sumSplitBefore
sumSplitFirst = Forall(
from proveit.logic import Forall, Or, Equals, Implies
from proveit.number import Reals
from proveit.number import LessThan, LessThanEquals, GreaterThan, GreaterThanEquals
from proveit.common import x, y, z
from proveit import beginAxioms, endAxioms
beginAxioms(locals())
lessThanEqualsDef = Forall([x, y],
Or(LessThan(x, y), Equals(x, y)),
domain=Reals,
conditions=LessThanEquals(x, y))
lessThanEqualsDef
greaterThanEqualsDef = Forall([x, y],
Or(GreaterThan(x, y), Equals(x, y)),
domain=Reals,
conditions=GreaterThanEquals(x, y))
greaterThanEqualsDef
reverseGreaterThanEquals = Forall((x, y),
Implies(GreaterThanEquals(x, y),
LessThanEquals(y, x)))
reverseGreaterThanEquals
reverseLessThanEquals = Forall((x, y),
Implies(LessThanEquals(x, y),
GreaterThanEquals(y, x)))
reverseLessThanEquals
reverseGreaterThan = Forall((x, y), Implies(GreaterThan(x, y), LessThan(y, x)))
intervalInInts = Forall((a, b), Forall(n, InSet(n, Integers), domain=Interval(a, b)), domain=Integers) intervalInInts intervalInNats = Forall((a, b), Forall(n, InSet(n, Naturals), domain=Interval(a, b)), domain=Naturals) intervalInNats intervalInNatsPos = Forall((a, b), Forall(n, InSet(n, NaturalsPos), domain=Interval(a, b)), domain=Integers, conditions=[GreaterThan(a, zero)]) intervalInNatsPos allInNegativeIntervalAreNegative = Forall((a, b), Forall(n, LessThan(n, zero), domain=Interval(a, b)), domain=Integers, conditions=[LessThan(b, zero)]) allInNegativeIntervalAreNegative allInPositiveIntervalArePositive = Forall((a, b), Forall(n, GreaterThan(n, zero), domain=Interval(a, b)), domain=Integers, conditions=[GreaterThan(a, zero)]) allInPositiveIntervalArePositive intervalLowerBound = Forall((a, b), Forall(n, LessThanEquals(a, n), domain=Interval(a, b)), domain=Integers) intervalLowerBound intervalUpperBound = Forall((a, b), Forall(n, LessThanEquals(n, b), domain=Interval(a, b)), domain=Integers) intervalUpperBound inInterval = Forall((a, b, n), InSet(n, Interval(a, b)), domain=Integers, conditions=[LessThanEquals(a, n), LessThanEquals(n, b)]) inInterval natsInInts = Forall(a,InSet(a,Integers),domain = Naturals) natsInInts natsInReals = Forall(a,InSet(a,Reals),domain = Naturals) natsInReals natsInComplexes = Forall(a,InSet(a,Complexes),domain = Naturals)
minRealPosClosure
maxRealClosure = Forall((a, b), InSet(Max(a, b), Reals), domain=Reals)
maxRealClosure
maxRealPosClosure = Forall((a, b), InSet(Max(a, b), RealsPos), domain=RealsPos)
maxRealPosClosure
relaxGreaterThan = Forall([a,b],
GreaterThanEquals(a,b),
domain = Reals,
conditions = GreaterThan(a,b))
relaxGreaterThan
relaxLessThan = Forall([a,b],
LessThanEquals(a,b),
domain = Reals,
conditions = LessThan(a,b))
relaxLessThan
lessThanInBools = Forall([a, b], InSet(LessThan(a, b), Booleans), domain=Reals)
lessThanInBools
lessThanEqualsInBools = Forall([a, b], InSet(LessThanEquals(a, b), Booleans), domain=Reals)
lessThanEqualsInBools
greaterThanInBools = Forall([a, b], InSet(GreaterThan(a, b), Booleans), domain=Reals)
greaterThanInBools
greaterThanEqualsInBools = Forall([a, b], InSet(GreaterThanEquals(a, b), Booleans), domain=Reals)
greaterThanEqualsInBools