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
Home
Index