Nuprl Lemma : unit-ball-to-unit-cube
∀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 < mdist(max-metric(n);λi.r0;p)} . g p ≡ (λp.(||p||/mdist(max-metric(n);p;λi.r0))*p) p)
∧ (g ∈ {p:ℝ^n| ||p|| ≤ r1} ⟶ {q:ℝ^n| mdist(max-metric(n);λi.r0;q) ≤ r1} )
∧ g:FUN(ℝ^n;ℝ^n)
∧ (∀x,y:{p:ℝ^n| ||p|| ≤ r1} . (mdist(max-metric(n);g x;g y) ≤ (r((2 * n) + 1) * mdist(rn-metric(n);x;y)))))
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,
rmul: a * b,
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],
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
real-vec: ℝ^n,
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
metric-leq: d1 ≤ d2,
rn-metric: rn-metric(n),
mdist: mdist(d;x;y),
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
scale-metric: c*d,
rless: x < y,
sq_exists: ∃x:A [B[x]],
guard: {T},
req_int_terms: t1 ≡ t2,
rneq: x ≠ y,
less_than': less_than'(a;b),
rdiv: (x/y),
rge: x ≥ y,
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
true: True,
meq: x ≡ y,
label: ...$L... t,
stable: Stable{P},
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
is-mfun: f:FUN(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,
rat_term_to_real: rat_term_to_real(f;t),
rtermMultiply: left "*" right,
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermSubtract: left "-" right,
rtermVar: rtermVar(var),
rtermConstant: "const",
pi1: fst(t),
pi2: snd(t),
sq_type: SQType(T),
respects-equality: respects-equality(S;T)
Lemmas referenced :
int-to-real_wf,
int_seg_wf,
max-metric-leq-rn-metric,
nat_plus_subtype_nat,
rn-metric-leq-max-metric,
nat_plus_wf,
real-vec-dist_wf,
nat_plus_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,
istype-void,
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-norm_wf,
real-vec-dist-from-zero,
req_functionality,
real-vec-dist-symmetry,
req_weakening,
mdist_wf,
real-vec_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,
rleq_functionality,
rless_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real-vec-mul_wf,
rdiv_wf,
remove-singularity-max-mfun,
mul_bounds_1a,
multiply_nat_wf,
rn-metric-complete,
rless-implies-rless,
rsub_wf,
rabs_wf,
mdist-symm,
mdist-rn-metric-mul,
rmul_functionality,
rabs-of-nonneg,
rmul_preserves_rleq,
rinv_wf2,
real-vec-norm-nonneg,
req_transitivity,
rmul-rinv,
req_inversion,
rmul-int,
rinv-mul-as-rdiv,
rmul_preserves_rleq2,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
sq_stable__less_than,
rleq_weakening,
squash_wf,
true_wf,
real_wf,
subtype_rel_self,
iff_weakening_equal,
rmul-rinv3,
rmul-nonneg-case1,
istype-false,
stable__rleq,
false_wf,
not_wf,
not-rless,
meq-max-metric,
rleq_antisymmetry,
mdist-nonneg,
meq-max-metric-iff-meq-rn-metric,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
mdist_functionality,
meq_weakening,
mdist-same,
mdist-max-metric-mul,
rleq_transitivity,
req-vec_wf,
subtype_rel_dep_function,
subtype_rel_sets_simple,
meq_wf,
real-vec-dist-identity,
req-vec_weakening,
meq-same,
req-vec_functionality,
real-vec-mul_functionality,
rdiv_functionality,
real-vec-norm_functionality,
rn-metric-meq,
real-vec-norm-diff-bound,
mdist-triangle-inequality,
metric-on-subtype,
radd_wf,
radd_functionality,
rabs-difference-bound-rleq,
rleq-implies-rleq,
radd_comm_eq,
itermAdd_wf,
real_term_value_add_lemma,
sq_stable__rleq,
r-triangle-inequality2,
rleq_weakening_rless,
rabs_functionality,
rsub-rdiv,
rneq_functionality,
rabs-rdiv,
assert-rat-term-eq2,
rtermSubtract_wf,
rtermDivide_wf,
rtermVar_wf,
rtermMultiply_wf,
rtermConstant_wf,
rabs-rmul,
rminus_wf,
itermMinus_wf,
real_term_value_minus_lemma,
rsub_functionality,
rmul-assoc,
rabs-difference-symmetry,
rmul_functionality_wrt_rleq2,
zero-rleq-rabs,
radd_functionality_wrt_rleq,
radd-int,
mdist-max-metric-mul2,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
rmul-distrib,
mdist-max-metric-mul-rleq,
req-vec_inversion,
radd-preserves-rleq,
radd-non-neg,
respects-equality-function,
respects-equality-set-trivial,
is-mfun_wf,
efficient-exp-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
lambdaEquality_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
universeIsType,
natural_numberEquality,
hypothesisEquality,
applyEquality,
because_Cache,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
dependent_set_memberEquality_alt,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
inrFormation_alt,
setIsType,
multiplyEquality,
closedConclusion,
functionIsType,
equalityIstype,
promote_hyp,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
universeEquality,
functionExtensionality,
setEquality,
unionEquality,
functionEquality,
unionIsType,
productIsType,
productEquality,
minusEquality,
inlFormation_alt,
cumulativity,
intEquality
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 < mdist(max-metric(n);\mlambda{}i.r0;p)\}
g p \mequiv{} (\mlambda{}p.(||p||/mdist(max-metric(n);p;\mlambda{}i.r0))*p) p)
\mwedge{} (g \mmember{} \{p:\mBbbR{}\^{}n| ||p|| \mleq{} r1\} {}\mrightarrow{} \{q:\mBbbR{}\^{}n| mdist(max-metric(n);\mlambda{}i.r0;q) \mleq{} r1\} )
\mwedge{} g:FUN(\mBbbR{}\^{}n;\mBbbR{}\^{}n)
\mwedge{} (\mforall{}x,y:\{p:\mBbbR{}\^{}n| ||p|| \mleq{} r1\} .
(mdist(max-metric(n);g x;g y) \mleq{} (r((2 * n) + 1) * mdist(rn-metric(n);x;y)))))
Date html generated:
2019_10_30-AM-11_25_49
Last ObjectModification:
2019_07_08-PM-04_55_28
Theory : real!vectors
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