Nuprl Lemma : dyadic-scaled-rationals-dense
∀a:ℝ. ((r0 < a) ⇒ dense-in-interval((-∞, ∞);λr.∃n:ℤ. ∃m:ℕ+. (r = (r(n) * (a/r(2^m))))))
Proof
Definitions occuring in Statement :
dense-in-interval: dense-in-interval(I;X),
riiint: (-∞, ∞),
rdiv: (x/y),
rless: x < y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
exp: i^n,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
dense-in-interval: dense-in-interval(I;X),
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
true: True,
exists: ∃x:A. B[x],
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int-to-real_wf,
riiint_wf,
i-member_wf,
real_wf,
set_wf,
rless_wf,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv3,
req_transitivity,
rless_functionality,
req-iff-rsub-is-0,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
rmul_wf,
rdiv_wf,
rmul_preserves_rless,
dyadic-rationals-dense,
true_wf,
member_riiint_lemma,
less_than_wf,
exp-positive,
rless-int,
nat_plus_subtype_nat,
exp_wf2,
rmul_preserves_req,
rmul_comm,
rsub_wf,
req_wf,
nat_plus_wf,
exists_wf,
rless-implies-rless,
req_weakening,
req_functionality
Rules used in proof :
natural_numberEquality,
lambdaEquality,
hypothesis,
hypothesisEquality,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
rename,
sqequalRule,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
productElimination,
independent_functionElimination,
inrFormation,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality,
baseClosed,
imageMemberEquality,
independent_pairFormation,
applyEquality,
productEquality,
dependent_pairFormation
Latex:
\mforall{}a:\mBbbR{}. ((r0 < a) {}\mRightarrow{} dense-in-interval((-\minfty{}, \minfty{});\mlambda{}r.\mexists{}n:\mBbbZ{}. \mexists{}m:\mBbbN{}\msupplus{}. (r = (r(n) * (a/r(2\^{}m))))))
Date html generated:
2017_10_03-AM-10_21_05
Last ObjectModification:
2017_08_01-PM-02_15_13
Theory : reals
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