Nuprl Lemma : metric-leq-cauchy
∀[X:Type]. ∀[d1,d2:metric(X)]. ∀c:{c:ℝ| r0 < c} . (d1 ≤ c*d2 ⇒ (∀x:ℕ ⟶ X. (mcauchy(d2;n.x n) ⇒ mcauchy(d1;n.x n))))
Proof
Definitions occuring in Statement :
mcauchy: mcauchy(d;n.x[n]),
metric-leq: d1 ≤ d2,
scale-metric: c*d,
metric: metric(X),
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
mcauchy: mcauchy(d;n.x[n]),
nat_plus: ℕ+,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
sq_exists: ∃x:A [B[x]],
metric-leq: d1 ≤ d2,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
metric: metric(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
scale-metric: c*d,
mdist: mdist(d;x;y),
sq_stable: SqStable(P),
squash: ↓T,
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top
Latex:
\mforall{}[X:Type]. \mforall{}[d1,d2:metric(X)].
\mforall{}c:\{c:\mBbbR{}| r0 < c\} . (d1 \mleq{} c*d2 {}\mRightarrow{} (\mforall{}x:\mBbbN{} {}\mrightarrow{} X. (mcauchy(d2;n.x n) {}\mRightarrow{} mcauchy(d1;n.x n))))
Date html generated:
2020_05_20-AM-11_57_12
Last ObjectModification:
2020_01_06-PM-00_18_14
Theory : reals
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