Nuprl Lemma : Cauchy-Schwarz3
∀[n:ℕ]. ∀[x,y:ℕn ⟶ ℝ].
(|Σ{x[i] * y[i] | 0≤i≤n - 1}| ≤ (rsqrt(Σ{x[i] * x[i] | 0≤i≤n - 1}) * rsqrt(Σ{y[i] * y[i] | 0≤i≤n - 1})))
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rsum: Σ{x[k] | n≤k≤m},
rleq: x ≤ y,
rabs: |x|,
rmul: a * b,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rleq: x ≤ y,
rnonneg: rnonneg(x),
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
less_than': less_than'(a;b),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
eq_int: (i =z j),
nequal: a ≠ b ∈ T
Lemmas referenced :
Cauchy-Schwarz2,
rsum_nonneg,
subtract_wf,
rmul_wf,
int_seg_wf,
subtract-add-cancel,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
square-nonneg,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
le_wf,
less_than'_wf,
rsub_wf,
rsum_wf,
nat_plus_properties,
int-to-real_wf,
nat_plus_wf,
real_wf,
nat_wf,
sq_stable__rleq,
rabs_wf,
rsqrt_wf,
rleq_wf,
req_wf,
equal_wf,
rnexp-rleq-iff,
zero-rleq-rabs,
rmul-nonneg-case1,
rsqrt_nonneg,
less_than_wf,
rnexp_wf,
false_wf,
uimplies_transitivity,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_functionality,
req_weakening,
req_transitivity,
rnexp-rmul,
rmul_functionality,
rsqrt-rnexp-2,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rnexp_unroll,
req_inversion,
rabs-rmul,
rabs-of-nonneg
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
natural_numberEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
addEquality,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
setEquality,
productEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbN{}n {}\mrightarrow{} \mBbbR{}].
(|\mSigma{}\{x[i] * y[i] | 0\mleq{}i\mleq{}n - 1\}| \mleq{} (rsqrt(\mSigma{}\{x[i] * x[i] | 0\mleq{}i\mleq{}n - 1\})
* rsqrt(\mSigma{}\{y[i] * y[i] | 0\mleq{}i\mleq{}n - 1\})))
Date html generated:
2017_10_03-AM-10_46_34
Last ObjectModification:
2017_07_28-AM-08_19_55
Theory : reals
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