Nuprl Lemma : q-Cauchy-Schwarz
∀[n:ℕ]. ∀[x,y:ℕn + 1 ⟶ ℚ].
((Σ0 ≤ i < n. x[i] * y[i] * Σ0 ≤ i < n. x[i] * y[i]) ≤ (Σ0 ≤ i < n. x[i] * x[i] * Σ0 ≤ i < n. y[i] * y[i]))
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
qmul: r * s,
rationals: ℚ,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
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,
subtype_rel: A ⊆r B,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
btrue: tt,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
le: A ≤ B,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
qmul_ident,
qmul_zero_qrng,
mon_ident_q,
qadd_preserves_qle,
q_distrib,
qadd_ac_1_q,
qadd_comm_q,
mon_assoc_q,
qinv_inv_q,
qmul_comm_qrng,
qmul_ac_1_qrng,
qmul_assoc_qrng,
qmul_over_minus_qrng,
qmul_over_plus_qrng,
qsub_wf,
q-square-non-neg,
le_wf,
int_formula_prop_le_lemma,
intformle_wf,
qsum_functionality_wrt_qle,
qsum-linearity2,
qsum-linearity1,
iff_weakening_equal,
qsum_product,
true_wf,
squash_wf,
qle_wf,
qadd_wf,
int_seg_properties,
nat_wf,
rationals_wf,
int_seg_wf,
lelt_wf,
qle_witness,
int-subtype-rationals,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
qsum_wf,
qmul_wf,
qmul_preserves_qle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
addEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
functionEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
lambdaFormation,
minusEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbQ{}].
((\mSigma{}0 \mleq{} i < n. x[i] * y[i] * \mSigma{}0 \mleq{} i < n. x[i] * y[i]) \mleq{} (\mSigma{}0 \mleq{} i < n. x[i] * x[i]
* \mSigma{}0 \mleq{} i < n. y[i] * y[i]))
Date html generated:
2016_05_15-PM-11_12_22
Last ObjectModification:
2016_01_16-PM-09_23_24
Theory : rationals
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