Nuprl Lemma : Cauchy-Schwarz-not-strict
∀[n:ℕ]. ∀[x,y:ℝ^n]. (¬(|x⋅y| < (||x|| * ||y||)) ⇐⇒ ∀i,j:ℕn. (((x j) * (y i)) = ((x i) * (y j))))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec: ℝ^n,
rless: x < y,
rabs: |x|,
req: x = y,
rmul: a * b,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
real-vec: ℝ^n,
so_apply: x[s],
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
int_seg_wf,
not_wf,
exists_wf,
rneq_wf,
rmul_wf,
all_wf,
req_wf,
Cauchy-Schwarz-strict,
rless_wf,
rabs_wf,
dot-product_wf,
real-vec-norm_wf,
iff_wf,
real-vec_wf,
nat_wf,
req_witness,
not-rneq,
rneq_functionality,
req_weakening,
nat_plus_properties,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
applyEquality,
independent_functionElimination,
voidElimination,
addLevel,
productElimination,
impliesFunctionality,
dependent_functionElimination,
impliesLevelFunctionality,
isect_memberFormation,
independent_pairEquality,
isect_memberEquality,
independent_isectElimination,
dependent_pairFormation,
unionElimination,
imageElimination,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (\mneg{}(|x\mcdot{}y| < (||x|| * ||y||)) \mLeftarrow{}{}\mRightarrow{} \mforall{}i,j:\mBbbN{}n. (((x j) * (y i)) = ((x i) * (y j))))
Date html generated:
2017_10_03-AM-10_53_00
Last ObjectModification:
2017_06_19-PM-04_20_29
Theory : reals
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