Nuprl Lemma : approx-ball-to-ball_wf
∀[n:ℕ]. ∀[k:ℕ+]. ∀[p:unit-ball-approx(n;k)].  (approx-ball-to-ball(k;p) ∈ B(n))
Proof
Definitions occuring in Statement : 
approx-ball-to-ball: approx-ball-to-ball(k;p)
, 
unit-ball-approx: unit-ball-approx(n;k)
, 
real-unit-ball: B(n)
, 
nat_plus: ℕ+
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
approx-ball-to-ball: approx-ball-to-ball(k;p)
, 
real-vec: ℝ^n
, 
subtype_rel: A ⊆r B
, 
unit-ball-approx: unit-ball-approx(n;k)
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
real-unit-ball: B(n)
, 
all: ∀x:A. B[x]
, 
prop: ℙ
, 
nat_plus: ℕ+
, 
ge: i ≥ j 
, 
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
, 
and: P ∧ Q
, 
iff: P 
⇐⇒ Q
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
dot-product: x⋅y
, 
rev_implies: P 
⇐ Q
, 
so_lambda: λ2x.t[x]
, 
lelt: i ≤ j < k
, 
less_than: a < b
, 
squash: ↓T
, 
so_apply: x[s]
, 
rneq: x ≠ y
, 
guard: {T}
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
pointwise-req: x[k] = y[k] for k ∈ [n,m]
, 
nequal: a ≠ b ∈ T 
, 
rat_term_to_real: rat_term_to_real(f;t)
, 
rtermMultiply: left "*" right
, 
rat_term_ind: rat_term_ind, 
rtermDivide: num "/" denom
, 
rtermVar: rtermVar(var)
, 
rtermConstant: "const"
, 
pi1: fst(t)
, 
true: True
, 
pi2: snd(t)
, 
sq_stable: SqStable(P)
, 
rdiv: (x/y)
, 
req_int_terms: t1 ≡ t2
Lemmas referenced : 
int-rdiv_wf, 
nat_plus_inc_int_nzero, 
int-to-real_wf, 
int_seg_wf, 
square-rleq-1-iff, 
real-vec-norm_wf, 
rleq_wf, 
unit-ball-approx_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, 
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, 
nat_plus_wf, 
istype-nat, 
rabs_wf, 
real-vec-norm-nonneg, 
rnexp_wf, 
dot-product_wf, 
iff_weakening_uiff, 
rleq_functionality, 
rabs-of-nonneg, 
req_weakening, 
real-vec-norm-squared, 
rsum_wf, 
subtract_wf, 
rmul_wf, 
int_seg_properties, 
decidable__lt, 
itermAdd_wf, 
itermSubtract_wf, 
int_term_value_add_lemma, 
int_term_value_subtract_lemma, 
istype-less_than, 
rdiv_wf, 
subtract-add-cancel, 
rless-int, 
rless_wf, 
rsum_functionality2, 
rmul_functionality, 
int-rdiv-req, 
sum_wf, 
rleq-int, 
rmul-int, 
rsum_int, 
rsum_functionality, 
mul_bounds_1b, 
rneq_functionality, 
rneq-int, 
int_entire_a, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
int_subtype_base, 
req_functionality, 
rmul-rdiv, 
mul_nat_plus, 
assert-rat-term-eq2, 
rtermDivide_wf, 
rtermVar_wf, 
rtermMultiply_wf, 
rtermConstant_wf, 
rdiv_functionality, 
rsum_linearity2, 
rmul_preserves_rleq, 
sq_stable__rless, 
rinv_wf2, 
itermMultiply_wf, 
req_transitivity, 
rmul-rinv3, 
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
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
sqequalRule, 
lambdaEquality_alt, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
applyEquality, 
hypothesis, 
setElimination, 
rename, 
inhabitedIsType, 
equalityTransitivity, 
equalitySymmetry, 
universeIsType, 
natural_numberEquality, 
dependent_set_memberEquality_alt, 
dependent_functionElimination, 
axiomEquality, 
unionElimination, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
dependent_pairFormation_alt, 
int_eqEquality, 
isect_memberEquality_alt, 
voidElimination, 
independent_pairFormation, 
isectIsTypeImplies, 
because_Cache, 
productElimination, 
lambdaFormation_alt, 
promote_hyp, 
imageElimination, 
productIsType, 
addEquality, 
closedConclusion, 
inrFormation_alt, 
multiplyEquality, 
equalityIstype, 
baseClosed, 
sqequalBase, 
applyLambdaEquality, 
imageMemberEquality
Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[k:\mBbbN{}\msupplus{}].  \mforall{}[p:unit-ball-approx(n;k)].    (approx-ball-to-ball(k;p)  \mmember{}  B(n))
Date html generated:
2019_10_30-AM-11_28_44
Last ObjectModification:
2019_06_28-PM-01_56_23
Theory : real!vectors
Home
Index