Nuprl Lemma : remove-singularity
∀[X:Type]. ∀[d:metric(X)].
∀k:ℕ. ∀f:{p:ℝ^k| r0 < ||p||} ⟶ X. ∀z:X.
((∃c:{c:ℝ| r0 ≤ c} . ∀m:ℕ+. ∀p:{p:ℝ^k| r0 < ||p||} . ((||p|| ≤ (r(4)/r(m))) ⇒ (mdist(d;f p;z) ≤ (c/r(m)))))
⇒ mcomplete(X with d)
⇒ (∃g:ℝ^k ⟶ X. ((∀p:ℝ^k. (req-vec(k;p;λi.r0) ⇒ g p ≡ z)) ∧ (∀p:{p:ℝ^k| r0 < ||p||} . g p ≡ f p))))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
mcomplete: mcomplete(M),
mk-metric-space: X with d,
mdist: mdist(d;x;y),
meq: x ≡ y,
metric: metric(X),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
real: ℝ,
eq_int: (i =z j),
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
mconverges-to: lim n→∞.x[n] = y,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
less_than: a < b,
sq_exists: ∃x:A [B[x]],
rless: x < y,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
incr-binary-seq: IBS,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
remove-singularity-seq: remove-singularity-seq(k;p;f;z),
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
ge: i ≥ j ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
real-vec: ℝ^n,
meq: x ≡ y,
uimplies: b supposing a,
cand: A c∧ B,
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
metric: metric(X),
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
ifthenelse_wf,
sq_stable__less_than,
ibs-property,
sq_stable__rless,
mdist-same,
rleq_functionality,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
istype-le,
rleq-int-fractions2,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
real-vec-norm-0,
real-vec-norm_functionality,
req_weakening,
rless_functionality,
int_term_value_add_lemma,
itermAdd_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
int_subtype_base,
lelt_wf,
set_subtype_base,
realvec-ibs-property,
assert_of_eq_int,
eqtt_to_assert,
realvec-ibs_wf,
eq_int_wf,
istype-universe,
metric_wf,
istype-nat,
mdist_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
nat_plus_properties,
rless-int,
rdiv_wf,
nat_plus_wf,
rleq_wf,
real_wf,
mk-metric-space_wf,
mcomplete_wf,
meq_wf,
real-vec-norm_wf,
rless_wf,
int_seg_wf,
req-vec_wf,
int-to-real_wf,
req_witness,
cauchy-mlimit-unique,
real-vec_wf,
mcauchy_wf,
remove-singularity-seq_wf,
cauchy-mlimit_wf,
remove-singularity-seq-mcauchy
Rules used in proof :
addEquality,
imageMemberEquality,
dependent_set_memberFormation_alt,
dependent_set_memberEquality_alt,
multiplyEquality,
imageElimination,
cumulativity,
promote_hyp,
sqequalBase,
baseClosed,
intEquality,
equalityIstype,
equalityElimination,
universeEquality,
instantiate,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
unionElimination,
dependent_functionElimination,
inrFormation_alt,
closedConclusion,
functionIsType,
productIsType,
setIsType,
independent_pairFormation,
productElimination,
natural_numberEquality,
because_Cache,
independent_isectElimination,
setElimination,
universeIsType,
inhabitedIsType,
isectIsType,
equalitySymmetry,
equalityTransitivity,
sqequalRule,
applyEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
rename,
independent_functionElimination,
lambdaFormation_alt,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)].
\mforall{}k:\mBbbN{}. \mforall{}f:\{p:\mBbbR{}\^{}k| r0 < ||p||\} {}\mrightarrow{} X. \mforall{}z:X.
((\mexists{}c:\{c:\mBbbR{}| r0 \mleq{} c\}
\mforall{}m:\mBbbN{}\msupplus{}. \mforall{}p:\{p:\mBbbR{}\^{}k| r0 < ||p||\} . ((||p|| \mleq{} (r(4)/r(m))) {}\mRightarrow{} (mdist(d;f p;z) \mleq{} (c/r(m)))))
{}\mRightarrow{} mcomplete(X with d)
{}\mRightarrow{} (\mexists{}g:\mBbbR{}\^{}k {}\mrightarrow{} X
((\mforall{}p:\mBbbR{}\^{}k. (req-vec(k;p;\mlambda{}i.r0) {}\mRightarrow{} g p \mequiv{} z)) \mwedge{} (\mforall{}p:\{p:\mBbbR{}\^{}k| r0 < ||p||\} . g p \mequiv{} f p))))
Date html generated:
2019_10_30-AM-11_24_44
Last ObjectModification:
2019_10_29-PM-01_33_50
Theory : real!vectors
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