Nuprl Lemma : remove-singularity-seq-mcauchy
∀[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)))))
⇒ (∀[p:ℝ^k]. mcauchy(d;n.remove-singularity-seq(k;p;f;z) n)))
Proof
Definitions occuring in Statement :
remove-singularity-seq: remove-singularity-seq(k;p;f;z),
real-vec-norm: ||x||,
real-vec: ℝ^n,
mcauchy: mcauchy(d;n.x[n]),
mdist: mdist(d;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,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
mcauchy: mcauchy(d;n.x[n]),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
sq_type: SQType(T),
rless: x < y,
sq_exists: ∃x:A [B[x]],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
remove-singularity-seq: remove-singularity-seq(k;p;f;z),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
incr-binary-seq: IBS,
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
rge: x ≥ y,
real: ℝ,
sq_stable: SqStable(P)
Lemmas referenced :
nat_plus_wf,
real-vec_wf,
real_wf,
rleq_wf,
int-to-real_wf,
rless_wf,
real-vec-norm_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,
mdist_wf,
istype-nat,
metric_wf,
istype-universe,
r-archimedean,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
r-archimedean-implies2,
rless-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
less_than_wf,
mul_nat_plus,
intformle_wf,
int_formula_prop_le_lemma,
istype-less_than,
rmul_preserves_rleq,
mul_bounds_1b,
rmul_wf,
rinv_wf2,
rneq_functionality,
rmul-int,
req_weakening,
int_entire_a,
itermSubtract_wf,
rleq_functionality,
req_transitivity,
rmul_functionality,
rinv_functionality2,
req_inversion,
rinv-of-rmul,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
nat_plus_subtype_nat,
realvec-ibs-property,
eq_int_wf,
realvec-ibs_wf,
eqtt_to_assert,
assert_of_eq_int,
int_seg_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lelt_wf,
istype-le,
remove-singularity-seq_wf,
rleq-int-fractions2,
decidable__le,
mdist-same,
int_seg_properties,
nequal_wf,
itermAdd_wf,
int_term_value_add_lemma,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq-int-fractions,
sq_stable__less_than,
rmul_preserves_rleq2,
sq_stable__rleq,
rinv-mul-as-rdiv,
mdist-symm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
rename,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
hypothesisEquality,
sqequalRule,
productIsType,
setIsType,
natural_numberEquality,
functionIsType,
setElimination,
closedConclusion,
because_Cache,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
applyEquality,
instantiate,
universeEquality,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
dependent_set_memberEquality_alt,
equalityIstype,
inhabitedIsType,
baseClosed,
sqequalBase,
equalityElimination,
promote_hyp,
applyLambdaEquality,
imageElimination,
addEquality,
imageMemberEquality
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{} (\mforall{}[p:\mBbbR{}\^{}k]. mcauchy(d;n.remove-singularity-seq(k;p;f;z) n)))
Date html generated:
2019_10_30-AM-10_16_23
Last ObjectModification:
2019_06_28-PM-01_55_57
Theory : real!vectors
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