Nuprl Lemma : interval-cube-uniform-continuity
∀I:{I:Interval| icompact(I)}
(iproper(I)
⇒ (∀n,k:ℕ. ∀f:{f:I^n ⟶ ℝ^k| ∀x,y:I^n. (req-vec(n;x;y)
⇒ req-vec(k;f x;f y))} . ∀e:{e:ℝ| r0 < e} .
∃d:ℕ+. ∀x,y:I^n. ((d(x;y) ≤ (r1/r(d)))
⇒ (d(f x;f y) ≤ e))))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y)
,
interval-vec: I^n
,
req-vec: req-vec(n;x;y)
,
real-vec: ℝ^n
,
icompact: icompact(I)
,
iproper: iproper(I)
,
interval: Interval
,
rdiv: (x/y)
,
rleq: x ≤ y
,
rless: x < y
,
int-to-real: r(n)
,
real: ℝ
,
nat_plus: ℕ+
,
nat: ℕ
,
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
Definitions unfolded in proof :
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
sq_stable: SqStable(P)
,
squash: ↓T
,
iproper: iproper(I)
,
ext-eq: A ≡ B
,
and: P ∧ Q
,
subtype_rel: A ⊆r B
,
real-cube: real-cube(n;a;b)
,
interval-vec: I^n
,
i-member: r ∈ I
,
rccint: [l, u]
,
nat: ℕ
,
real-vec: ℝ^n
,
prop: ℙ
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
guard: {T}
,
istype: istype(T)
,
exists: ∃x:A. B[x]
,
nat_plus: ℕ+
,
rneq: x ≠ y
,
or: P ∨ Q
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
rless: x < y
,
sq_exists: ∃x:A [B[x]]
,
ge: i ≥ j
,
decidable: Dec(P)
,
not: ¬A
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
false: False
,
top: Top
,
icompact: icompact(I)
Lemmas referenced :
icompact-is-rccint,
sq_stable__icompact,
int_seg_wf,
i-member_wf,
real-cube_wf,
left-endpoint_wf,
right-endpoint_wf,
rccint_wf,
interval-vec_wf,
real-cube-uniform-continuity,
subtype_rel_sets,
real-vec_wf,
all_wf,
req-vec_wf,
subtype_rel_set,
subtype_rel_dep_function,
subtype_rel_weakening,
ext-eq_inversion,
rleq_wf,
real-vec-dist_wf,
rdiv_wf,
int-to-real_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,
rless_wf,
real_wf,
istype-nat,
iproper_wf,
interval_wf,
icompact_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
productElimination,
universeIsType,
natural_numberEquality,
functionIsType,
applyEquality,
functionEquality,
inhabitedIsType,
dependent_pairFormation_alt,
closedConclusion,
inrFormation_alt,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
setIsType
Latex:
\mforall{}I:\{I:Interval| icompact(I)\}
(iproper(I)
{}\mRightarrow{} (\mforall{}n,k:\mBbbN{}. \mforall{}f:\{f:I\^{}n {}\mrightarrow{} \mBbbR{}\^{}k| \mforall{}x,y:I\^{}n. (req-vec(n;x;y) {}\mRightarrow{} req-vec(k;f x;f y))\} .
\mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}d:\mBbbN{}\msupplus{}. \mforall{}x,y:I\^{}n. ((d(x;y) \mleq{} (r1/r(d))) {}\mRightarrow{} (d(f x;f y) \mleq{} e))))
Date html generated:
2019_10_30-AM-10_14_36
Last ObjectModification:
2019_06_28-PM-01_52_04
Theory : real!vectors
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