Nuprl Lemma : metric-leq-complete
∀[X:Type]. ∀[d1,d2:metric(X)].
(d2 ≤ d1
⇒ (∀x:ℕ ⟶ X. (mcauchy(d2;n.x n) ⇒ (∃y:ℕ ⟶ X. (subsequence(a,b.a ≡ b;n.x n;n.y n) ∧ mcauchy(d1;n.y n)))))
⇒ mcomplete(X with d1)
⇒ mcomplete(X with d2))
Proof
Definitions occuring in Statement :
mcomplete: mcomplete(M),
mcauchy: mcauchy(d;n.x[n]),
mk-metric-space: X with d,
metric-leq: d1 ≤ d2,
meq: x ≡ y,
metric: metric(X),
subsequence: subsequence(a,b.E[a; b];m.x[m];n.y[n]),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
mcomplete: mcomplete(M),
mk-metric-space: X with d,
all: ∀x:A. B[x],
metric: metric(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
mconverges: x[n]↓ as n→∞,
mconverges-to: lim n→∞.x[n] = y,
mcauchy: mcauchy(d;n.x[n]),
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
sq_exists: ∃x:A [B[x]],
nat: ℕ,
ge: i ≥ j ,
subsequence: subsequence(a,b.E[a; b];m.x[m];n.y[n]),
guard: {T},
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
metric-leq: d1 ≤ d2
Latex:
\mforall{}[X:Type]. \mforall{}[d1,d2:metric(X)].
(d2 \mleq{} d1
{}\mRightarrow{} (\mforall{}x:\mBbbN{} {}\mrightarrow{} X
(mcauchy(d2;n.x n)
{}\mRightarrow{} (\mexists{}y:\mBbbN{} {}\mrightarrow{} X. (subsequence(a,b.a \mequiv{} b;n.x n;n.y n) \mwedge{} mcauchy(d1;n.y n)))))
{}\mRightarrow{} mcomplete(X with d1)
{}\mRightarrow{} mcomplete(X with d2))
Date html generated:
2020_05_20-AM-11_59_24
Last ObjectModification:
2019_12_14-PM-04_49_25
Theory : reals
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