Nuprl Lemma : find-approx-fixpoint-unit-ball
∀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 :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
real-unit-ball: B(n),
subtype_rel: A ⊆r B,
not: ¬A,
false: False
Lemmas referenced :
find-approx-fp_wf,
real_wf,
rless_wf,
int-to-real_wf,
real-unit-ball_wf,
real-vec-dist_wf,
real-vec-sep_wf,
istype-void,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
setIsType,
universeIsType,
isectElimination,
natural_numberEquality,
functionIsType,
because_Cache,
sqequalRule,
productIsType,
setElimination,
rename,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry
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_25
Last ObjectModification:
2019_07_30-PM-00_54_30
Theory : real!vectors
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