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
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