Nuprl Lemma : implies-real-vec-norm-rleq
∀[n:ℕ]. ∀[x:ℝ^n]. ∀[c:ℝ]. ((∀i:ℕn. (|x i| ≤ c)) ⇒ (||x|| ≤ (rsqrt(r(n)) * c)))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
real-vec: ℝ^n,
rsqrt: rsqrt(x),
rleq: x ≤ y,
rabs: |x|,
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,
all: ∀x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
real-vec: ℝ^n,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
so_apply: x[s],
top: Top,
so_lambda: λ2x.t[x],
subtract: n - m,
dot-product: x⋅y,
real-vec-norm: ||x||,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rge: x ≥ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbR{}\^{}n]. \mforall{}[c:\mBbbR{}]. ((\mforall{}i:\mBbbN{}n. (|x i| \mleq{} c)) {}\mRightarrow{} (||x|| \mleq{} (rsqrt(r(n)) * c)))
Date html generated:
2020_05_20-PM-00_37_09
Last ObjectModification:
2020_01_06-PM-00_14_28
Theory : reals
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