Nuprl Lemma : real-ball-uniform-continuity
∀k:ℕ. ∀n:ℕ+. ∀f:{f:B(n;r1) ⟶ ℝ^k| ∀x,y:B(n;r1). (req-vec(n;x;y) ⇒ req-vec(k;f x;f y))} . ∀e:{e:ℝ| r0 < e} .
∃d:ℕ+. ∀x,y:B(n;r1). ((d(x;y) ≤ (r1/r(d))) ⇒ (d(f x;f y) ≤ e))
Proof
Definitions occuring in Statement :
real-ball: B(n;r),
real-vec-dist: d(x;y),
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
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],
member: t ∈ T,
homeomorphic+: homeomorphic+(X;dX;Y;dY),
exists: ∃x:A. B[x],
and: P ∧ Q,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
iproper: iproper(I),
top: Top,
less_than: a < b,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
real-ball: B(n;r),
mfun: FUN(X ⟶ Y),
compose: f o g,
uiff: uiff(P;Q),
is-mfun: f:FUN(X;Y),
so_apply: x[s],
meq: x ≡ y,
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
rsqrt: rsqrt(x),
rroot: rroot(i;x),
ifthenelse: if b then t else f fi ,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m,
btrue: tt,
rroot-abs: rroot-abs(i;x),
fastexp: i^n,
efficient-exp-ext,
genrec: genrec,
subtract: n - m,
rn-metric: rn-metric(n),
interval-vec: I^n,
mdist: mdist(d;x;y),
rneq: x ≠ y,
guard: {T},
metric-leq: d1 ≤ d2,
scale-metric: c*d,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
sq_stable: SqStable(P)
Lemmas referenced :
unit-balls-homeomorphic+-2,
interval-cube-uniform-continuity,
rccint-icompact,
int-to-real_wf,
rleq-int,
istype-false,
rccint_wf,
icompact_wf,
left_endpoint_rccint_lemma,
istype-void,
right_endpoint_rccint_lemma,
rless-int,
i-finite_wf,
nat_plus_subtype_nat,
real_wf,
rless_wf,
real-ball_wf,
nat_plus_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
real-vec_wf,
req-vec_wf,
nat_plus_wf,
istype-nat,
compose_wf,
interval-vec_wf,
meq-max-metric,
real-vec-dist-identity,
mul_nat_plus,
rleq_wf,
real-vec-dist_wf,
rdiv_wf,
multiply_nat_plus,
decidable__lt,
itermMultiply_wf,
intformeq_wf,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
rn-metric-leq-max-metric,
rmul_wf,
mdist_wf,
max-metric_wf,
metric-on-subtype,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rmul_preserves_rleq2,
itermSubtract_wf,
rleq_functionality,
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,
rmul_preserves_rleq,
rinv_wf2,
req_transitivity,
rmul-rinv,
req_weakening,
req_functionality,
rmul_functionality,
req_inversion,
rmul-int,
sq_stable__rleq,
rleq_weakening,
real-vec-dist_functionality,
rn-metric-meq,
req-vec_weakening,
req-vec_functionality,
efficient-exp-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
minusEquality,
natural_numberEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
independent_pairFormation,
because_Cache,
dependent_set_memberEquality_alt,
universeIsType,
isect_memberEquality_alt,
voidElimination,
imageMemberEquality,
baseClosed,
applyEquality,
setIsType,
functionIsType,
setElimination,
rename,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
multiplyEquality,
inrFormation_alt,
applyLambdaEquality,
equalityIstype,
promote_hyp,
imageElimination
Latex:
\mforall{}k:\mBbbN{}. \mforall{}n:\mBbbN{}\msupplus{}. \mforall{}f:\{f:B(n;r1) {}\mrightarrow{} \mBbbR{}\^{}k| \mforall{}x,y:B(n;r1). (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:B(n;r1). ((d(x;y) \mleq{} (r1/r(d))) {}\mRightarrow{} (d(f x;f y) \mleq{} e))
Date html generated:
2019_10_30-AM-11_27_05
Last ObjectModification:
2019_07_08-PM-05_42_26
Theory : real!vectors
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