Nuprl Lemma : remove-singularity-max
∀[X:Type]. ∀[d:metric(X)].
∀k:ℕ. ∀f:{p:ℝ^k| r0 < mdist(max-metric(k);p;λi.r0)} ⟶ X. ∀z:X.
((∃c:{c:ℝ| r0 ≤ c}
∀m:ℕ+. ∀p:{p:ℝ^k| r0 < mdist(max-metric(k);p;λi.r0)} .
((mdist(max-metric(k);p;λi.r0) ≤ (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 < mdist(max-metric(k);p;λi.r0)} . g p ≡ f p))))
Proof
Definitions occuring in Statement :
max-metric: max-metric(n),
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
metric: metric(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
cand: A c∧ B,
uimplies: b supposing a,
meq: x ≡ y,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
remove-singularity-max-seq: remove-singularity-max-seq(k;p;f;z),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
incr-binary-seq: IBS,
ifthenelse: if b then t else f fi ,
mdist: mdist(d;x;y),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
mconverges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:A [B[x]],
less_than': less_than'(a;b),
true: True,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
rless: x < y,
eq_int: (i =z j),
real: ℝ
Lemmas referenced :
remove-singularity-max-seq-mcauchy,
int-to-real_wf,
int_seg_wf,
cauchy-mlimit_wf,
remove-singularity-max-seq_wf,
mcauchy_wf,
cauchy-mlimit-unique,
req_witness,
req-vec_wf,
rless_wf,
mdist_wf,
real-vec_wf,
max-metric_wf,
meq_wf,
mcomplete_wf,
mk-metric-space_wf,
real_wf,
rleq_wf,
nat_plus_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-nat,
metric_wf,
istype-universe,
eq_int_wf,
realvec-max-ibs_wf,
eqtt_to_assert,
assert_of_eq_int,
realvec-max-ibs-property,
set_subtype_base,
lelt_wf,
int_subtype_base,
req-vec-meq-max-metric,
rless_transitivity1,
rleq_weakening,
rless_irreflexivity,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
istype-le,
rleq-int-fractions2,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
rleq_functionality,
mdist-same,
req_weakening,
sq_stable__rless,
ibs-property,
sq_stable__less_than,
ifthenelse_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
productElimination,
universeIsType,
natural_numberEquality,
promote_hyp,
independent_functionElimination,
dependent_pairFormation_alt,
applyEquality,
equalityTransitivity,
equalitySymmetry,
isectIsType,
inhabitedIsType,
independent_isectElimination,
because_Cache,
independent_pairFormation,
setIsType,
productIsType,
functionIsType,
closedConclusion,
inrFormation_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
instantiate,
universeEquality,
equalityElimination,
equalityIstype,
intEquality,
baseClosed,
sqequalBase,
cumulativity,
dependent_set_memberFormation_alt,
dependent_set_memberEquality_alt,
multiplyEquality,
imageElimination,
imageMemberEquality,
addEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)].
\mforall{}k:\mBbbN{}. \mforall{}f:\{p:\mBbbR{}\^{}k| r0 < mdist(max-metric(k);p;\mlambda{}i.r0)\} {}\mrightarrow{} X. \mforall{}z:X.
((\mexists{}c:\{c:\mBbbR{}| r0 \mleq{} c\}
\mforall{}m:\mBbbN{}\msupplus{}. \mforall{}p:\{p:\mBbbR{}\^{}k| r0 < mdist(max-metric(k);p;\mlambda{}i.r0)\} .
((mdist(max-metric(k);p;\mlambda{}i.r0) \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 < mdist(max-metric(k);p;\mlambda{}i.r0)\} . g p \mequiv{} f p))))
Date html generated:
2019_10_30-AM-11_24_52
Last ObjectModification:
2019_07_03-PM-04_53_36
Theory : real!vectors
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