Nuprl Lemma : Cauchy-Schwarz-equality
∀[n:ℕ]. ∀[x,y:ℝ^n]. ((r0 < ||y||) ⇒ (|x⋅y| = (||x|| * ||y||)) ⇒ req-vec(n;x;(x⋅y/||y||^2)*y))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec-mul: a*X,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rdiv: (x/y),
rless: x < y,
rabs: |x|,
rnexp: x^k1,
req: x = y,
rmul: a * b,
int-to-real: r(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
subtype_rel: A ⊆r B,
real-vec-mul: a*X,
exists: ∃x:A. B[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top,
dot-product: x⋅y,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
pointwise-req: x[k] = y[k] for k ∈ [n,m]
Lemmas referenced :
rnexp-positive,
real-vec-norm_wf,
false_wf,
le_wf,
req_inversion,
rabs_wf,
dot-product_wf,
rmul_wf,
rless_transitivity1,
rleq_weakening,
rless_irreflexivity,
rless_wf,
Cauchy-Schwarz-not-strict,
rnexp_wf,
int-to-real_wf,
req_wf,
req_witness,
real-vec-mul_wf,
rdiv_wf,
int_seg_wf,
real-vec_wf,
nat_wf,
real-vec-norm-positive-iff,
rneq-symmetry,
rmul_preserves_req,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
itermConstant_wf,
req_weakening,
req_functionality,
req_transitivity,
rmul_functionality,
rmul-rinv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmul_assoc,
rmul-one,
rmul_comm,
rinv-mul-as-rdiv,
rmul-rinv3,
real-vec-norm-squared,
rsum_functionality,
subtract_wf,
subtract-add-cancel,
int_seg_properties,
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
dot-product-linearity2,
intformle_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
independent_isectElimination,
because_Cache,
voidElimination,
productElimination,
inrFormation,
isect_memberFormation,
lambdaEquality,
applyEquality,
setElimination,
rename,
isect_memberEquality,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality,
unionElimination,
dependent_pairFormation,
addEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. ((r0 < ||y||) {}\mRightarrow{} (|x\mcdot{}y| = (||x|| * ||y||)) {}\mRightarrow{} req-vec(n;x;(x\mcdot{}y/||y||\^{}2)*y))
Date html generated:
2017_10_03-AM-10_53_25
Last ObjectModification:
2017_06_19-PM-04_23_26
Theory : reals
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