Nuprl Lemma : close-reals-iff
∀[x,y:ℝ]. ∀[k:ℕ+]. uiff(|x - y| ≤ (r1/r(k));∀m:ℕ+. ((|(x m) - y m| * k) ≤ ((4 * k) + (2 * m))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
real: ℝ,
rnonneg: rnonneg(x),
rleq: x ≤ y,
prop: ℙ,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
less_than': less_than'(a;b),
rational-approx: (x within 1/n),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
squash: ↓T,
less_than: a < b,
sq_exists: ∃x:A [B[x]],
rless: x < y,
nat: ℕ,
sq_stable: SqStable(P),
int-rdiv: (a)/k1,
has-value: (a)↓,
int-to-real: r(n),
rsub: x - y,
rabs: |x|,
rminus: -(x),
absval: |i|,
ge: i ≥ j ,
subtract: n - m
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}]. uiff(|x - y| \mleq{} (r1/r(k));\mforall{}m:\mBbbN{}\msupplus{}. ((|(x m) - y m| * k) \mleq{} ((4 * k) + (2 * m))))
Date html generated:
2020_05_20-AM-11_05_00
Last ObjectModification:
2019_12_28-PM-08_14_32
Theory : reals
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