Nuprl Lemma : approx-fixpoint-unit-ball-0-ext
∀n:ℕ+. ∀f:{f:B(n) ⟶ B(n)|
(∀e:{e:ℝ| r0 < e} . ∃del:{del:ℝ| r0 < del} . ∀x,y:B(n). ((d(x;y) < del) ⇒ (d(f x;f y) < e)))
∧ (¬(∀x:B(n). f x ≠ x))} . ∀e:{e:ℝ| r0 < e} .
∃t:B(n) ⟶ 𝔹
((∀p:B(n). ((↑(t p)) ⇒ (↓d(f p;p) < e))) ∧ (↓∃k:ℕ+. ∃q:unit-ball-approx(n;k). (↑(t approx-ball-to-ball(k;q)))))
Proof
Definitions occuring in Statement :
approx-ball-to-ball: approx-ball-to-ball(k;p),
unit-ball-approx: unit-ball-approx(n;k),
real-unit-ball: B(n),
real-vec-sep: a ≠ b,
real-vec-dist: d(x;y),
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
assert: ↑b,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
approx-fixpoint-unit-ball-0,
rless-cases,
rmul_preserves_rless,
rless-int,
rless_functionality,
radd-preserves-rless,
sq_stable__rless,
rmul_functionality_wrt_rless,
rless-iff4,
rless-iff-large-diff,
radd_functionality_wrt_rless2,
rinv-positive,
regular-less-iff,
decidable__le,
any: any x
Lemmas referenced :
approx-fixpoint-unit-ball-0,
rless-cases,
rmul_preserves_rless,
rless-int,
rless_functionality,
radd-preserves-rless,
sq_stable__rless,
rmul_functionality_wrt_rless,
rless-iff4,
rless-iff-large-diff,
radd_functionality_wrt_rless2,
rinv-positive,
regular-less-iff,
decidable__le
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}f:\{f:B(n) {}\mrightarrow{} B(n)|
(\mforall{}e:\{e:\mBbbR{}| r0 < e\}
\mexists{}del:\{del:\mBbbR{}| r0 < del\} . \mforall{}x,y:B(n). ((d(x;y) < del) {}\mRightarrow{} (d(f x;f y) < e)))
\mwedge{} (\mneg{}(\mforall{}x:B(n). f x \mneq{} x))\} . \mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}t:B(n) {}\mrightarrow{} \mBbbB{}
((\mforall{}p:B(n). ((\muparrow{}(t p)) {}\mRightarrow{} (\mdownarrow{}d(f p;p) < e)))
\mwedge{} (\mdownarrow{}\mexists{}k:\mBbbN{}\msupplus{}. \mexists{}q:unit-ball-approx(n;k). (\muparrow{}(t approx-ball-to-ball(k;q)))))
Date html generated:
2019_10_30-AM-11_29_10
Last ObjectModification:
2019_07_30-PM-00_31_32
Theory : real!vectors
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