Nuprl Lemma : rclose-or-sep_wf
∀[K:ℕ+]. ∀[x,y:ℝ].
(rclose-or-sep(K;x;y) ∈ {i:ℕ3|
((i = 1 ∈ ℤ) ⇒ ((r1/r(K)) < (y - x)))
∧ ((i = 2 ∈ ℤ) ⇒ ((r1/r(K)) < (x - y)))
∧ ((i = 0 ∈ ℤ) ⇒ (|x - y| < (r(2)/r(K))))} )
Proof
Definitions occuring in Statement :
rclose-or-sep: rclose-or-sep(K;x;y),
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+,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rclose-or-sep: rclose-or-sep(K;x;y),
reals-close-or-rneq-ext,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T
Latex:
\mforall{}[K:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbR{}].
(rclose-or-sep(K;x;y) \mmember{} \{i:\mBbbN{}3|
((i = 1) {}\mRightarrow{} ((r1/r(K)) < (y - x)))
\mwedge{} ((i = 2) {}\mRightarrow{} ((r1/r(K)) < (x - y)))
\mwedge{} ((i = 0) {}\mRightarrow{} (|x - y| < (r(2)/r(K))))\} )
Date html generated:
2020_05_20-AM-11_06_13
Last ObjectModification:
2019_11_06-PM-05_17_58
Theory : reals
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