Nuprl Lemma : mtb-cantor-map-continuous
∀[X:Type]. ∀[d:metric(X)]. ∀[cmplt:mcomplete(X with d)]. ∀[mtb:m-TB(X;d)]. ∀[k:ℕ+]. ∀[p,q:mtb-cantor(mtb)].
mdist(d;mtb-cantor-map(d;cmplt;mtb;p);mtb-cantor-map(d;cmplt;mtb;q)) ≤ (r1/r(k))
supposing ∀i:ℕ. ((i ≤ (18 * k)) ⇒ ((p i) = (q i) ∈ ℤ))
Proof
Definitions occuring in Statement :
mtb-cantor-map: mtb-cantor-map(d;cmplt;mtb;p),
mtb-cantor: mtb-cantor(mtb),
m-TB: m-TB(X;d),
mcomplete: mcomplete(M),
mk-metric-space: X with d,
mdist: mdist(d;x;y),
metric: metric(X),
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
multiply: n * m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
mtb-cantor-map: mtb-cantor-map(d;cmplt;mtb;p),
all: ∀x:A. B[x],
guard: {T},
implies: P ⇒ Q,
metric: metric(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
nat: ℕ,
nat_plus: ℕ+,
mtb-cantor: mtb-cantor(mtb),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
m-TB: m-TB(X;d),
pi1: fst(t),
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
mtb-seq: mtb-seq(mtb;s),
spreadn: spread3,
lelt: i ≤ j < k,
less_than: a < b,
less_than': less_than'(a;b),
sq_type: SQType(T),
true: True,
mcauchy: mcauchy(d;n.x[n]),
sq_exists: ∃x:A [B[x]],
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
sq_stable: SqStable(P),
mconverges-to: lim n→∞.x[n] = y,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
uiff: uiff(P;Q),
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q)
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[cmplt:mcomplete(X with d)]. \mforall{}[mtb:m-TB(X;d)]. \mforall{}[k:\mBbbN{}\msupplus{}].
\mforall{}[p,q:mtb-cantor(mtb)].
mdist(d;mtb-cantor-map(d;cmplt;mtb;p);mtb-cantor-map(d;cmplt;mtb;q)) \mleq{} (r1/r(k))
supposing \mforall{}i:\mBbbN{}. ((i \mleq{} (18 * k)) {}\mRightarrow{} ((p i) = (q i)))
Date html generated:
2020_05_20-PM-00_00_34
Last ObjectModification:
2019_12_28-PM-11_48_47
Theory : reals
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