Nuprl Lemma : hyp-distance-lemma1
∀[rv:InnerProductSpace]. ∀[x,y:Point]. (r1 ≤ ((rsqrt(r1 + x^2) * rsqrt(r1 + y^2)) - x ⋅ y))
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rsqrt: rsqrt(x),
rleq: x ≤ y,
rsub: x - y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
ss-point: Point,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
cand: A c∧ B,
or: P ∨ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
guard: {T},
req_int_terms: t1 ≡ t2,
top: Top,
nat: ℕ,
rge: x ≥ y
Lemmas referenced :
radd-preserves-rleq,
int-to-real_wf,
rsub_wf,
rmul_wf,
rsqrt_wf,
rv-ip_wf,
rleq_functionality,
radd_wf,
radd-non-neg,
rleq-int,
false_wf,
rv-ip-nonneg,
rleq_wf,
real_wf,
req_wf,
radd_comm,
rsqrt_nonneg,
rmul-nonneg,
less_than'_wf,
nat_plus_wf,
ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
square-rleq-implies,
rnexp_wf,
le_wf,
req_weakening,
req_transitivity,
rnexp-rmul,
rmul_functionality,
rsqrt-rnexp-2,
rv-Cauchy-Schwarz,
itermConstant_wf,
rnexp2,
radd_functionality,
req_inversion,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
rleq-implies-rleq,
rv-sub_wf,
rv-ip-sub-squared
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
because_Cache,
applyEquality,
sqequalRule,
hypothesisEquality,
productElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
inlFormation,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
isect_memberEquality,
voidElimination,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x,y:Point]. (r1 \mleq{} ((rsqrt(r1 + x\^{}2) * rsqrt(r1 + y\^{}2)) - x \mcdot{} y))
Date html generated:
2017_10_05-AM-00_29_05
Last ObjectModification:
2017_06_23-PM-05_48_26
Theory : inner!product!spaces
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