Nuprl Lemma : mdist-max-metric-ub
∀[n:ℕ]. ∀[x,y:ℝ^n]. ∀i:ℕn. (|(x i) - y i| ≤ mdist(max-metric(n);x;y))
Proof
Definitions occuring in Statement :
max-metric: max-metric(n),
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
natural_number: $n
Definitions unfolded in proof :
max-metric: max-metric(n),
mdist: mdist(d;x;y),
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
decidable: Dec(P),
real-vec: ℝ^n,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. \mforall{}i:\mBbbN{}n. (|(x i) - y i| \mleq{} mdist(max-metric(n);x;y))
Date html generated:
2020_05_20-PM-00_42_21
Last ObjectModification:
2019_11_12-AM-11_17_22
Theory : reals
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