Nuprl Lemma : ravg-dist-when-rleq
∀[x,y:ℝ]. ((ravg(x;y) - x) = ((r1/r(2)) * (y - x))) ∧ ((y - ravg(x;y)) = ((r1/r(2)) * (y - x))) supposing x ≤ y
Proof
Definitions occuring in Statement :
ravg: ravg(x;y),
rdiv: (x/y),
rleq: x ≤ y,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
ravg-dist,
ravg-weak-between,
req_witness,
rsub_wf,
ravg_wf,
rmul_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rleq_wf,
real_wf,
rabs_wf,
rleq-implies-rleq,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_functionality,
rabs-of-nonneg,
rmul_functionality,
req_weakening,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_var_lemma,
real_term_value_const_lemma,
rabs-difference-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
because_Cache,
independent_functionElimination,
hypothesis,
independent_pairFormation,
sqequalRule,
independent_pairEquality,
isectElimination,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
inrFormation_alt,
imageMemberEquality,
baseClosed,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
voidElimination
Latex:
\mforall{}[x,y:\mBbbR{}].
((ravg(x;y) - x) = ((r1/r(2)) * (y - x))) \mwedge{} ((y - ravg(x;y)) = ((r1/r(2)) * (y - x)))
supposing x \mleq{} y
Date html generated:
2019_10_29-AM-10_03_39
Last ObjectModification:
2019_01_11-AM-11_10_53
Theory : reals
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