Nuprl Lemma : union-metric-space-complete
∀T:Type. ∀eq:EqDecider(T). ∀X:T ⟶ MetricSpace. ((∀i:T. mcomplete(X i)) ⇒ mcomplete(union-metric-space(T;eq;X)))
Proof
Definitions occuring in Statement :
mcomplete: mcomplete(M),
union-metric-space: union-metric-space(T;eq;X),
metric-space: MetricSpace,
deq: EqDecider(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
union-metric-space: union-metric-space(T;eq;X),
mcomplete: mcomplete(M),
mk-metric-space: X with d,
mcauchy: mcauchy(d;n.x[n]),
member: t ∈ T,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
sq_exists: ∃x:A [B[x]],
metric-space: MetricSpace,
pi1: fst(t),
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
eqof: eqof(d),
pi2: snd(t),
subtype_rel: A ⊆r B,
bfalse: ff,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
mdist: mdist(d;x;y),
metric: metric(X),
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
label: ...$L... t,
rless: x < y,
cand: A c∧ B,
mconverges: x[n]↓ as n→∞,
mconverges-to: lim n→∞.x[n] = y
Lemmas referenced :
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
mcauchy_wf,
eqtt_to_assert,
safe-assert-deq,
rmin_wf,
mdist_wf,
subtype_rel-equal,
metric_wf,
int-to-real_wf,
istype-universe,
istype-nat,
mcomplete_wf,
metric-space_wf,
deq_wf,
nat_properties,
decidable__le,
intformand_wf,
intformle_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
istype-le,
rleq_wf,
rdiv_wf,
rless-int,
rless_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal_wf,
rmul_preserves_rleq2,
rleq-int,
istype-false,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
rinv_wf2,
rleq_functionality,
req_transitivity,
rmul-int,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
nat_plus_properties,
imax_wf,
subtract_wf,
imax_ub,
nat_plus_wf,
ifthenelse_wf,
le_int_wf,
assert_of_le_int,
int_term_value_subtract_lemma,
le_wf,
le_functionality,
le_weakening,
add_functionality_wrt_le,
squash_wf,
true_wf,
add_functionality_wrt_eq,
imax_unfold,
subtype_rel_self,
iff_weakening_equal,
pi1_wf_top,
istype-top,
subtype_rel_product,
top_wf,
real_wf,
rless-cases,
rless-int-fractions,
int_term_value_mul_lemma,
rleq_weakening_rless,
rmin_strict_ub,
rless-int-fractions3,
rless_transitivity1,
rless_irreflexivity,
trivial-int-eq1,
rmin_lb,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
less_than_wf,
int_subtype_base,
mconverges-to_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality_alt,
natural_numberEquality,
introduction,
extract_by_obid,
unionElimination,
isectElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
hypothesisEquality,
setElimination,
rename,
productEquality,
applyEquality,
inhabitedIsType,
productElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
because_Cache,
independent_pairFormation,
productIsType,
applyLambdaEquality,
closedConclusion,
functionIsType,
instantiate,
universeEquality,
addEquality,
int_eqEquality,
inrFormation_alt,
imageMemberEquality,
baseClosed,
promote_hyp,
cumulativity,
hyp_replacement,
dependent_set_memberFormation_alt,
inlFormation_alt,
intEquality,
imageElimination,
multiplyEquality,
dependent_pairEquality_alt,
sqequalBase,
spreadEquality
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}X:T {}\mrightarrow{} MetricSpace.
((\mforall{}i:T. mcomplete(X i)) {}\mRightarrow{} mcomplete(union-metric-space(T;eq;X)))
Date html generated:
2019_10_30-AM-06_46_35
Last ObjectModification:
2019_10_02-AM-10_58_02
Theory : reals
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