Nuprl Lemma : approx-fixpoint-unit-ball-1-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} .
∃p:B(n). (↓d(f p;p) < e)
Proof
Definitions occuring in Statement :
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: ℕ,
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,
it: ⋅,
approx-ball-to-ball: approx-ball-to-ball(k;p),
int-rdiv: (a)/k1,
outl: outl(x),
bottom: ⊥,
let: let,
approx-fixpoint-unit-ball-1,
decidable__equal_int,
approx-fixpoint-unit-ball-0-ext,
sq_stable__ex_nat_plus,
decidable__exists-unit-ball-approx,
decidable__assert,
decidable__int_equal,
sq_stable__ex_int_upper,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
approx-fixpoint-unit-ball-1,
lifting-strict-int_eq,
istype-void,
strict4-decide,
lifting-strict-decide,
strict4-spread,
decidable__equal_int,
approx-fixpoint-unit-ball-0-ext,
sq_stable__ex_nat_plus,
decidable__exists-unit-ball-approx,
decidable__assert,
decidable__int_equal,
sq_stable__ex_int_upper
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}n:\mBbbN{}. \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{}p:B(n). (\mdownarrow{}d(f p;p) < e)
Date html generated:
2019_10_30-AM-11_29_16
Last ObjectModification:
2019_07_30-PM-00_49_17
Theory : real!vectors
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