Nuprl Lemma : implies-real-vec-dist-rleq
∀[n:ℕ]. ∀[x,y:ℝ^n]. ∀[c:ℝ]. ((∀i:ℕn. (|(x i) - y i| ≤ c)) ⇒ (d(x;y) ≤ (rsqrt(r(n)) * c)))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec: ℝ^n,
rsqrt: rsqrt(x),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
real-vec-dist: d(x;y),
all: ∀x:A. B[x],
real-vec-sub: X - Y,
rev_uimplies: rev_uimplies(P;Q),
real-vec: ℝ^n,
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
nat: ℕ,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
implies-real-vec-norm-rleq,
real-vec-sub_wf,
rleq_functionality_wrt_implies,
rabs_wf,
rsub_wf,
rleq_weakening_equal,
rleq_weakening,
int_seg_wf,
rleq_wf,
le_witness_for_triv,
real_wf,
real-vec_wf,
istype-nat,
itermSubtract_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
applyEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
functionIsType,
universeIsType,
natural_numberEquality,
setElimination,
rename,
lambdaEquality_alt,
productElimination,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
approximateComputation,
int_eqEquality,
voidElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. \mforall{}[c:\mBbbR{}]. ((\mforall{}i:\mBbbN{}n. (|(x i) - y i| \mleq{} c)) {}\mRightarrow{} (d(x;y) \mleq{} (rsqrt(r(n)) * c)))
Date html generated:
2019_10_30-AM-08_30_01
Last ObjectModification:
2019_06_18-PM-01_55_58
Theory : reals
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