Nuprl Lemma : approx-fixpoint-unit-ball-0
∀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 :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
real-unit-ball: B(n),
isr: isr(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
sq_stable: SqStable(P),
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
real-vec-sep: a ≠ b,
rge: x ≥ y,
stable: Stable{P}
Lemmas referenced :
rmul_preserves_rless,
rdiv_wf,
rless-int,
rless-cases,
rmul_wf,
int-to-real_wf,
real-vec-dist_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
real-unit-ball_wf,
istype-true,
bfalse_wf,
btrue_wf,
nat_plus_subtype_nat,
istype-assert,
squash_wf,
rless_wf,
nat_plus_wf,
unit-ball-approx_wf,
assert_wf,
approx-ball-to-ball_wf,
real_wf,
real-vec-sep_wf,
itermSubtract_wf,
itermMultiply_wf,
rinv_wf2,
sq_stable__rless,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
nequal_wf,
radd-preserves-rless,
radd_wf,
itermAdd_wf,
rless_functionality,
req_transitivity,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
int-rinv-cancel,
real_term_value_add_lemma,
small-reciprocal-real,
rmin_wf,
rmin_strict_ub,
rmin-rleq,
rless_transitivity1,
decidable__lt,
real-unit-ball-totally-bounded1,
mul_nat_plus,
istype-less_than,
decidable__exists-unit-ball-approx,
multiply_nat_wf,
decidable__assert,
isr_wf,
rneq-int,
rless_transitivity2,
rsub_wf,
rless-implies-rless,
rleq_weakening,
rleq_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
stable__rleq,
false_wf,
not_wf,
not-rless,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
real-vec-triangle-inequality,
real-vec-dist-symmetry,
req_weakening,
rleq_weakening_rless,
radd_functionality_wrt_rless2,
radd_functionality_wrt_rleq,
radd_functionality_wrt_rless1,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
isectElimination,
independent_isectElimination,
sqequalRule,
hypothesis,
inrFormation_alt,
productElimination,
independent_functionElimination,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
closedConclusion,
dependent_set_memberEquality_alt,
setElimination,
rename,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
applyEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
equalityIstype,
functionIsType,
productIsType,
productEquality,
setIsType,
imageElimination,
instantiate,
cumulativity,
intEquality,
sqequalBase,
minusEquality,
multiplyEquality,
unionEquality,
functionEquality,
unionIsType
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_07
Last ObjectModification:
2019_07_30-PM-00_30_01
Theory : real!vectors
Home
Index