Nuprl Lemma : dyadic-rationals-dense
dense-in-interval((-∞, ∞);λr.∃n:ℤ. ∃m:ℕ+. (r = (r(n)/r(2^m))))
Proof
Definitions occuring in Statement :
dense-in-interval: dense-in-interval(I;X),
riiint: (-∞, ∞),
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
exp: i^n,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
lambda: λx.A[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
dense-in-interval: dense-in-interval(I;X),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
exists: ∃x:A. B[x],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
false: False,
uiff: uiff(P;Q),
nat: ℕ,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
int_upper: {i...},
subtype_rel: A ⊆r B,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
real: ℝ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
rational-approx: (x within 1/n),
subtract: n - m
Latex:
dense-in-interval((-\minfty{}, \minfty{});\mlambda{}r.\mexists{}n:\mBbbZ{}. \mexists{}m:\mBbbN{}\msupplus{}. (r = (r(n)/r(2\^{}m))))
Date html generated:
2020_05_20-PM-00_12_07
Last ObjectModification:
2020_01_06-PM-00_23_01
Theory : reals
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