Nuprl Lemma : rational-lower-approx-as-rat2real
∀x:ℝ. ∀n:ℕ+. ∃q:ℚ. ((below x within 1/n) = rat2real(q))
Proof
Definitions occuring in Statement :
rational-lower-approx: (below x within 1/n),
rat2real: rat2real(q),
req: x = y,
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
rationals: ℚ
Definitions unfolded in proof :
all: ∀x:A. B[x],
rational-lower-approx: (below x within 1/n),
has-value: (a)↓,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat_plus: ℕ+,
real: ℝ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rat2real: rat2real(q),
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. \mexists{}q:\mBbbQ{}. ((below x within 1/n) = rat2real(q))
Date html generated:
2020_05_20-AM-11_03_53
Last ObjectModification:
2019_11_21-AM-11_06_03
Theory : reals
Home
Index