Nuprl Lemma : unit-cube-to-unit-ball
∀n:ℕ+
∃g:ℝ^n ⟶ ℝ^n
((∀p:ℝ^n. (req-vec(n;p;λi.r0) ⇒ g p ≡ λi.r0))
∧ (∀p:{p:ℝ^n| r0 < ||p||} . g p ≡ (λp.(mdist(max-metric(n);λi.r0;p)/||p||)*p) p)
∧ (g ∈ {q:ℝ^n| mdist(max-metric(n);λi.r0;q) ≤ r1} ⟶ {q:ℝ^n| mdist(rn-metric(n);λi.r0;q) ≤ r1} )
∧ g:FUN(ℝ^n;ℝ^n))
Proof
Definitions occuring in Statement :
max-metric: max-metric(n),
rn-metric: rn-metric(n),
real-vec-norm: ||x||,
real-vec-mul: a*X,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
is-mfun: f:FUN(X;Y),
mdist: mdist(d;x;y),
meq: x ≡ y,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
subtract: n - m,
genrec: genrec,
efficient-exp-ext,
fastexp: i^n,
rroot-abs: rroot-abs(i;x),
btrue: tt,
remainder: n rem m,
modulus: a mod n,
eq_int: (i =z j),
isEven: isEven(n),
ifthenelse: if b then t else f fi ,
rroot: rroot(i;x),
rsqrt: rsqrt(x),
real-vec-norm: ||x||,
real-vec-dist: d(x;y),
meq: x ≡ y,
is-mfun: f:FUN(X;Y),
respects-equality: respects-equality(S;T),
so_apply: x[s],
so_lambda: λ2x.t[x],
cand: A c∧ B,
stable: Stable{P},
true: True,
rdiv: (x/y),
real: ℝ,
squash: ↓T,
sq_stable: SqStable(P),
less_than': less_than'(a;b),
rneq: x ≠ y,
req_int_terms: t1 ≡ t2,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
scale-metric: c*d,
rless: x < y,
sq_exists: ∃x:A [B[x]],
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
mdist: mdist(d;x;y),
rn-metric: rn-metric(n),
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
real-vec: ℝ^n,
metric-leq: d1 ≤ d2,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
meq-same,
req-vec_weakening,
real-vec-norm_functionality,
rdiv_functionality,
real-vec-mul_functionality,
req-vec_functionality,
real-vec-dist-identity,
metric-on-subtype,
rn-metric-meq,
meq-max-metric,
is-mfun_wf,
respects-equality-set-trivial,
respects-equality-function,
subtype_rel_dep_function,
meq_wf,
mdist-same,
mdist-rn-metric-mul,
meq_weakening,
mdist_functionality,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
real-vec-norm-nonneg,
rleq_antisymmetry,
real-vec-norm-is-0,
not-rless,
not_wf,
false_wf,
stable__rleq,
req-vec_wf,
meq-max-metric-iff-meq-rn-metric,
rinv-mul-as-rdiv,
rabs_functionality,
mdist-nonneg,
mdist-symm,
squash_wf,
true_wf,
real_wf,
subtype_rel_self,
iff_weakening_equal,
rmul-int,
rmul-rinv,
rmul_functionality,
rmul-rinv3,
req_transitivity,
rabs-of-nonneg,
rmul_preserves_rleq,
rinv_wf2,
rleq_weakening_rless,
rmul_preserves_rleq2,
sq_stable__less_than,
rleq_transitivity,
mdist-max-metric-mul,
rabs_wf,
sq_stable__rless,
max-metric-complete,
istype-false,
remove-singularity-mfun,
rdiv_wf,
real-vec-mul_wf,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
rless_functionality,
rleq_functionality,
mdist_wf,
rn-metric_wf,
scale-metric_wf,
rleq-int,
rleq_wf,
max-metric_wf,
rmul_preserves_rless,
rless-int,
decidable__lt,
rless_wf,
rless_transitivity1,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
req_weakening,
real-vec-dist-symmetry,
req_functionality,
real-vec-dist-from-zero,
real-vec-norm_wf,
real-vec-dist_wf,
istype-le,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_plus_properties,
real-vec_wf,
int_seg_wf,
int-to-real_wf,
nat_plus_wf,
rn-metric-leq-max-metric,
nat_plus_subtype_nat,
max-metric-leq-rn-metric,
efficient-exp-ext
Rules used in proof :
productIsType,
unionIsType,
functionEquality,
unionEquality,
setEquality,
functionExtensionality,
instantiate,
universeEquality,
addEquality,
promote_hyp,
equalityIstype,
imageElimination,
baseClosed,
imageMemberEquality,
functionIsType,
closedConclusion,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
setIsType,
inrFormation_alt,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
productElimination,
rename,
setElimination,
lambdaEquality_alt,
dependent_functionElimination,
universeIsType,
because_Cache,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}\msupplus{}
\mexists{}g:\mBbbR{}\^{}n {}\mrightarrow{} \mBbbR{}\^{}n
((\mforall{}p:\mBbbR{}\^{}n. (req-vec(n;p;\mlambda{}i.r0) {}\mRightarrow{} g p \mequiv{} \mlambda{}i.r0))
\mwedge{} (\mforall{}p:\{p:\mBbbR{}\^{}n| r0 < ||p||\} . g p \mequiv{} (\mlambda{}p.(mdist(max-metric(n);\mlambda{}i.r0;p)/||p||)*p) p)
\mwedge{} (g \mmember{} \{q:\mBbbR{}\^{}n| mdist(max-metric(n);\mlambda{}i.r0;q) \mleq{} r1\} {}\mrightarrow{} \{q:\mBbbR{}\^{}n| mdist(rn-metric(n);\mlambda{}i.r0;q) \mleq{} r1\} )
\mwedge{} g:FUN(\mBbbR{}\^{}n;\mBbbR{}\^{}n))
Date html generated:
2019_10_30-AM-11_25_27
Last ObjectModification:
2019_10_29-PM-01_08_34
Theory : real!vectors
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