Nuprl Lemma : equiv-metrics-preserve-complete
∀[X:Type]. ∀[d1,d2:metric(X)].
((∃c1,c2:{s:ℝ| r0 < s} . (c1*d1 ≤ d2 ∧ c2*d2 ≤ d1)) ⇒ (mcomplete(X with d1) ⇐⇒ mcomplete(X with d2)))
Proof
Definitions occuring in Statement :
mcomplete: mcomplete(M),
mk-metric-space: X with d,
metric-leq: d1 ≤ d2,
scale-metric: c*d,
metric: metric(X),
rless: x < y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
sq_stable: SqStable(P),
rev_implies: P ⇐ Q,
or: P ∨ Q,
cand: A c∧ B,
prop: ℙ,
squash: ↓T,
uimplies: b supposing a,
rneq: x ≠ y,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
less_than: a < b,
less_than': less_than'(a;b),
true: True,
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
subsequence: subsequence(a,b.E[a; b];m.x[m];n.y[n]),
nat: ℕ,
le: A ≤ B,
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
metric: metric(X),
metric-leq: d1 ≤ d2,
scale-metric: c*d,
mdist: mdist(d;x;y),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
sq_stable__rless,
int-to-real_wf,
rmul_wf,
rmul-is-positive,
rless_wf,
rmul_preserves_rless,
rdiv_wf,
mcomplete_wf,
mk-metric-space_wf,
real_wf,
metric-leq_wf,
scale-metric_wf,
subtype_rel_sets_simple,
rleq_wf,
rleq_weakening_rless,
metric_wf,
istype-universe,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
rinv_wf2,
rless-int,
rless_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rinv-of-rmul,
rmul-rinv,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
metric-leq-complete,
istype-le,
le_witness_for_triv,
nat_properties,
sq_stable__less_than,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
meq-same,
meq_wf,
metric-leq-cauchy,
subsequence_wf,
istype-nat,
mcauchy_wf,
scale-metric-complete,
sq_stable__rleq,
mdist_wf,
rmul_preserves_rleq,
rleq_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
setElimination,
rename,
introduction,
extract_by_obid,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_functionElimination,
inlFormation_alt,
sqequalRule,
productIsType,
universeIsType,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
independent_isectElimination,
inrFormation_alt,
setIsType,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
instantiate,
universeEquality,
closedConclusion,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
addEquality,
unionElimination,
functionIsType,
isectIsType,
equalityIstype
Latex:
\mforall{}[X:Type]. \mforall{}[d1,d2:metric(X)].
((\mexists{}c1,c2:\{s:\mBbbR{}| r0 < s\} . (c1*d1 \mleq{} d2 \mwedge{} c2*d2 \mleq{} d1))
{}\mRightarrow{} (mcomplete(X with d1) \mLeftarrow{}{}\mRightarrow{} mcomplete(X with d2)))
Date html generated:
2019_10_30-AM-06_49_26
Last ObjectModification:
2019_10_02-AM-11_00_11
Theory : reals
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