Nuprl Lemma : ravg-dist
∀x,y:ℝ. ((|ravg(x;y) - x| = ((r1/r(2)) * |y - x|)) ∧ (|ravg(x;y) - y| = ((r1/r(2)) * |y - x|)))
Proof
Definitions occuring in Statement :
ravg: ravg(x;y),
rdiv: (x/y),
rabs: |x|,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
top: Top,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
ravg: ravg(x;y),
not: ¬A,
false: False,
le: A ≤ B,
nat: ℕ,
subtype_rel: A ⊆r B,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
prop: ℙ,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
and: P ∧ Q,
all: ∀x:A. B[x]
Lemmas referenced :
radd_functionality,
rabs-rmul,
req_inversion,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_minus_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv3,
rabs_functionality,
rmul_functionality,
req_transitivity,
req_functionality,
req_weakening,
le_wf,
false_wf,
absval_pos,
equal_wf,
nat_wf,
absval_wf,
req-int,
iff_weakening_equal,
rabs-int,
true_wf,
squash_wf,
req_wf,
itermConstant_wf,
itermMultiply_wf,
rinv_wf2,
req-iff-rsub-is-0,
itermMinus_wf,
itermAdd_wf,
itermVar_wf,
itermSubtract_wf,
rmul_comm,
rless_wf,
rminus_wf,
radd_wf,
int-to-real_wf,
ravg_wf,
rsub_wf,
rabs_wf,
real_wf,
rless-int,
rdiv_wf,
rmul_wf,
rmul_preserves_req,
rabs-rminus
Rules used in proof :
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
approximateComputation,
minusEquality,
dependent_set_memberEquality,
intEquality,
rename,
setElimination,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
hypothesisEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
dependent_functionElimination,
inrFormation,
hypothesis,
sqequalRule,
independent_isectElimination,
because_Cache,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x,y:\mBbbR{}. ((|ravg(x;y) - x| = ((r1/r(2)) * |y - x|)) \mwedge{} (|ravg(x;y) - y| = ((r1/r(2)) * |y - x|)))
Date html generated:
2017_10_03-AM-08_42_11
Last ObjectModification:
2017_07_31-AM-10_29_42
Theory : reals
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