Nuprl Lemma : Cauchy-Schwarz2-strict
∀n:ℕ. ∀x,y:ℕn ⟶ ℝ.
(∃i,j:ℕn. x[j] * y[i] ≠ x[i] * y[j]
⇐⇒ (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},
rneq: x ≠ y,
rless: x < y,
rmul: a * b,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
top: Top,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
subtract: n - m,
true: True,
and: P ∧ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
nat_wf,
subtract_wf,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
exists_wf,
int_seg_wf,
rneq_wf,
rmul_wf,
nat_plus_properties,
itermAdd_wf,
int_term_value_add_lemma,
rless_wf,
int-to-real_wf,
real_wf,
rsum-empty,
Cauchy-Schwarz1-strict-iff,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
subtract-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
applyEquality,
functionExtensionality,
imageElimination,
functionEquality,
dependent_set_memberEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbN{}n {}\mrightarrow{} \mBbbR{}.
(\mexists{}i,j:\mBbbN{}n. x[j] * y[i] \mneq{} x[i] * y[j]
\mLeftarrow{}{}\mRightarrow{} (\mSigma{}\{x[i] * y[i] | 0\mleq{}i\mleq{}n - 1\} * \mSigma{}\{x[i] * y[i] | 0\mleq{}i\mleq{}n - 1\}) < (\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:
2017_10_03-AM-09_04_40
Last ObjectModification:
2017_06_19-PM-04_00_20
Theory : reals
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