Nuprl Lemma : Cauchy-Schwarz1
∀[n:ℕ]. ∀[x,y:ℕn + 1 ⟶ ℝ].
((Σ{x[i] * y[i] | 0≤i≤n} * Σ{x[i] * y[i] | 0≤i≤n}) ≤ (Σ{x[i] * x[i] | 0≤i≤n} * Σ{y[i] * y[i] | 0≤i≤n}))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rleq: x ≤ y,
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],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
guard: {T},
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m]
Lemmas referenced :
rmul_preserves_rleq,
rmul_wf,
rsum_wf,
int_seg_wf,
rless-int,
less_than'_wf,
rsub_wf,
nat_plus_wf,
real_wf,
nat_wf,
int-to-real_wf,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
le_wf,
rleq_functionality,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul_functionality,
req_weakening,
rsum_functionality2,
radd_wf,
rsum_product,
radd_functionality,
int_seg_properties,
req_inversion,
rsum_linearity2,
rsum_linearity1,
rsum_functionality_wrt_rleq,
square-nonneg,
rleq-implies-rleq,
real_term_value_add_lemma,
rleq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
addEquality,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberEquality,
voidElimination,
dependent_set_memberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
lambdaFormation,
addLevel,
impliesFunctionality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}].
((\mSigma{}\{x[i] * y[i] | 0\mleq{}i\mleq{}n\} * \mSigma{}\{x[i] * y[i] | 0\mleq{}i\mleq{}n\}) \mleq{} (\mSigma{}\{x[i] * x[i] | 0\mleq{}i\mleq{}n\}
* \mSigma{}\{y[i] * y[i] | 0\mleq{}i\mleq{}n\}))
Date html generated:
2017_10_03-AM-09_03_56
Last ObjectModification:
2017_07_28-AM-07_41_04
Theory : reals
Home
Index