Nuprl Lemma : reals-close-or-rneq
∀K:ℕ+
(∃g:ℝ ⟶ ℝ ⟶ ℕ3 [(∀x,y:ℝ.
((((g x y) = 1 ∈ ℤ) ⇒ ((r1/r(K)) < (y - x)))
∧ (((g x y) = 2 ∈ ℤ) ⇒ ((r1/r(K)) < (x - y)))
∧ (((g x y) = 0 ∈ ℤ) ⇒ (|x - y| < (r(2)/r(K))))))])
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
int_seg: {i..j-},
lelt: i ≤ j < k,
cand: A c∧ B,
sq_stable: SqStable(P),
sq_type: SQType(T),
true: True
Latex:
\mforall{}K:\mBbbN{}\msupplus{}
(\mexists{}g:\mBbbR{} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbN{}3 [(\mforall{}x,y:\mBbbR{}.
((((g x y) = 1) {}\mRightarrow{} ((r1/r(K)) < (y - x)))
\mwedge{} (((g x y) = 2) {}\mRightarrow{} ((r1/r(K)) < (x - y)))
\mwedge{} (((g x y) = 0) {}\mRightarrow{} (|x - y| < (r(2)/r(K))))))])
Date html generated:
2020_05_20-AM-11_05_37
Last ObjectModification:
2019_12_28-PM-08_15_07
Theory : reals
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