Nuprl Lemma : Cauchy-Schwarz1-strict-iff
∀n:ℕ. ∀x,y:ℕn + 1 ⟶ ℝ.
(∃i,j:ℕn + 1. x[j] * y[i] ≠ x[i] * y[j]
⇐⇒ (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},
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],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
le: A ≤ B,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
rev_uimplies: rev_uimplies(P;Q),
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rneq: x ≠ y
Lemmas referenced :
Cauchy-Schwarz1-strict,
exists_wf,
int_seg_wf,
rneq_wf,
rmul_wf,
rmul_preserves_rless,
rsum_wf,
int-to-real_wf,
rless-int,
rless_functionality,
rmul_comm,
rless_wf,
real_wf,
nat_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
int_seg_properties,
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_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,
rmul_functionality,
rsum_product,
radd_functionality,
req_transitivity,
req_inversion,
rsum_linearity2,
rsum_functionality2,
rsum_linearity1,
rless-implies-rless,
rsub_wf,
rsum_linearity-rsub,
rsum_functionality,
rnexp_wf,
false_wf,
req_functionality,
rnexp2,
rsum-of-nonneg-positive-iff,
rsum_nonneg,
rnexp2-nonneg,
equal_wf,
rneq-iff-rabs,
rabs_wf,
rabs-positive-iff,
rmul-is-positive
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_pairFormation,
independent_functionElimination,
isectElimination,
natural_numberEquality,
addEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
productElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
functionEquality,
addLevel,
impliesFunctionality,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
unionElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
inrFormation,
inlFormation
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}.
(\mexists{}i,j:\mBbbN{}n + 1. x[j] * y[i] \mneq{} x[i] * y[j]
\mLeftarrow{}{}\mRightarrow{} (\mSigma{}\{x[i] * y[i] | 0\mleq{}i\mleq{}n\} * \mSigma{}\{x[i] * y[i] | 0\mleq{}i\mleq{}n\}) < (\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_04_29
Last ObjectModification:
2017_06_19-PM-03_54_47
Theory : reals
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