Nuprl Lemma : Cauchy-Schwarz2
∀[n:ℕ]. ∀[x,y:ℕn ⟶ ℝ].
((Σ{x[i] * y[i] | 0≤i≤n - 1} * Σ{x[i] * y[i] | 0≤i≤n - 1}) ≤ (Σ{x[i] * x[i] | 0≤i≤n - 1}
* Σ{y[i] * y[i] | 0≤i≤n - 1}))
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],
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,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
nat_plus: ℕ+,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
subtract: n - m,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
le_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
intformeq_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
decidable__le,
subtract_wf,
Cauchy-Schwarz1,
rsum-empty,
int-to-real_wf,
rleq_weakening_equal,
nat_wf,
real_wf,
nat_plus_wf,
int_seg_wf,
lelt_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
nat_plus_properties,
subtract-add-cancel,
rsum_wf,
rmul_wf,
rsub_wf,
less_than'_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
hypothesis,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
lambdaEquality,
productElimination,
independent_pairEquality,
applyEquality,
dependent_set_memberEquality,
independent_pairFormation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
addEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbN{}n {}\mrightarrow{} \mBbbR{}].
((\mSigma{}\{x[i] * y[i] | 0\mleq{}i\mleq{}n - 1\} * \mSigma{}\{x[i] * y[i] | 0\mleq{}i\mleq{}n - 1\}) \mleq{} (\mSigma{}\{x[i] * x[i] | 0\mleq{}i\mleq{}n - 1\}
* \mSigma{}\{y[i] * y[i] | 0\mleq{}i\mleq{}n - 1\}))
Date html generated:
2016_05_18-AM-07_51_47
Last ObjectModification:
2016_01_17-AM-02_17_28
Theory : reals
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