Nuprl Lemma : unit-ball-approxn
∀[k:ℕ]. ∀[n:ℕ+].
unit-ball-approx(n;k) ≡ {f:ℕn ⟶ {-k..k + 1-}|
∃q:unit-ball-approx(n - 1;k)
∃z:{-k..k + 1-}
(((Σ((q i) * (q i) | i < n - 1) + (z * z)) ≤ (k * k))
∧ (∀i:ℕn - 1. ((f i) = (q i) ∈ ℤ))
∧ ((f (n - 1)) = z ∈ ℤ))}
Proof
Definitions occuring in Statement :
unit-ball-approx: unit-ball-approx(n;k),
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
unit-ball-approx: unit-ball-approx(n;k),
exists: ∃x:A. B[x],
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than': less_than'(a;b),
cand: A c∧ B,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
true: True,
guard: {T},
sq_type: SQType(T)
Lemmas referenced :
unit-ball-approx_wf,
subtract_wf,
nat_plus_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
int_seg_wf,
sum_wf,
int_seg_properties,
int_seg_subtype,
istype-false,
multiply-is-int-iff,
set_subtype_base,
le_wf,
int_subtype_base,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
lelt_wf,
decidable__lt,
istype-less_than,
nat_plus_wf,
istype-nat,
squash_wf,
true_wf,
sum_split1,
subtype_rel_self,
iff_weakening_equal,
sum-unroll,
istype-top,
square_non_neg,
le_functionality,
add_functionality_wrt_le,
le_weakening,
subtype_rel_function,
add-is-int-iff,
itermMultiply_wf,
itermAdd_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
false_wf,
subtype_base_sq,
add_functionality_wrt_eq,
sum_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesisEquality,
sqequalRule,
productIsType,
universeIsType,
extract_by_obid,
isectElimination,
hypothesis,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
minusEquality,
addEquality,
because_Cache,
productElimination,
imageElimination,
multiplyEquality,
applyEquality,
functionIsType,
equalityIstype,
lambdaFormation_alt,
baseApply,
closedConclusion,
baseClosed,
intEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalBase,
setIsType,
independent_pairEquality,
axiomEquality,
isectIsTypeImplies,
imageMemberEquality,
instantiate,
universeEquality,
lessCases,
axiomSqEquality,
pointwiseFunctionality,
promote_hyp,
cumulativity
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}].
unit-ball-approx(n;k) \mequiv{} \{f:\mBbbN{}n {}\mrightarrow{} \{-k..k + 1\msupminus{}\}|
\mexists{}q:unit-ball-approx(n - 1;k)
\mexists{}z:\{-k..k + 1\msupminus{}\}
(((\mSigma{}((q i) * (q i) | i < n - 1) + (z * z)) \mleq{} (k * k))
\mwedge{} (\mforall{}i:\mBbbN{}n - 1. ((f i) = (q i)))
\mwedge{} ((f (n - 1)) = z))\}
Date html generated:
2019_10_30-AM-11_28_17
Last ObjectModification:
2019_06_28-PM-01_56_16
Theory : real!vectors
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