Nuprl Lemma : mcomplete-stable-union
∀[X:Type]. ∀[d:metric(X)]. ∀[T:Type]. ∀[P:T ⟶ X ⟶ ℙ].
finite(T) ⇒ mcomplete(X with d) ⇒ (∀i:T. mcompact({x:X| P[i;x]} ;d)) ⇒ mcomplete(stable-union(X;T;i,x.P[i;x]) with \000Cd)
supposing ∀i:T. ∀x,y:X. (P[i;x] ⇒ y ≡ x ⇒ P[i;y])
Proof
Definitions occuring in Statement :
mcompact: mcompact(X;d),
mcomplete: mcomplete(M),
mk-metric-space: X with d,
meq: x ≡ y,
metric: metric(X),
finite: finite(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
m-closed-subspace: m-closed-subspace(X;d;A),
mcompact: mcompact(X;d),
rless: x < y,
sq_exists: ∃x:A [B[x]],
mconverges-to: lim n→∞.x[n] = y,
rge: x ≥ y,
surject: Surj(A;B;f),
biject: Bij(A;B;f),
rneq: x ≠ y,
sq_type: SQType(T),
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q),
true: True,
subtract: n - m,
uiff: uiff(P;Q),
cand: A c∧ B,
less_than': less_than'(a;b),
decidable: Dec(P),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
pi1: fst(t),
squash: ↓T,
less_than: a < b,
guard: {T},
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
nat: ℕ,
equipollent: A ~ B,
finite: finite(T),
iff: P ⇐⇒ Q,
istype: istype(T),
and: P ∧ Q,
metric: metric(X),
so_lambda: λ2x y.t[x; y],
false: False,
not: ¬A,
or: P ∨ Q,
prop: ℙ,
so_apply: x[s],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
mconverges: x[n]↓ as n→∞,
stable-union: Error :stable-union,
subtype_rel: A ⊆r B,
member: t ∈ T,
all: ∀x:A. B[x],
mk-metric-space: X with d,
mcomplete: mcomplete(M),
implies: P ⇒ Q,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rleq_weakening_rless,
set-metric-subspace,
m-closed-iff-complete,
compact-dist-zero,
rleq_antisymmetry,
compact-dist-nonneg,
not-rless,
mdist-symm,
req_weakening,
rleq_functionality,
rless_irreflexivity,
rless_transitivity1,
rless-int-fractions,
int_term_value_add_lemma,
itermAdd_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
rleq-int-fractions,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
iff_weakening_equal,
true_wf,
squash_wf,
equal_wf,
subtype_rel_sets_simple,
nat_properties,
rless-int,
decidable__equal_int,
lelt_wf,
subtype_base_sq,
le_weakening,
le_functionality,
imax_ub,
le_witness_for_triv,
int_formula_prop_eq_lemma,
intformeq_wf,
imax_nat_plus,
int_term_value_subtract_lemma,
itermSubtract_wf,
imax_wf,
le-add-cancel2,
add-commutes,
add-zero,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
add-associates,
condition-implies-le,
not-le-2,
decidable__le,
istype-false,
int_seg_subtype,
subtype_rel_function,
nat_plus_properties,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
le_wf,
primrec-wf2,
istype-less_than,
istype-le,
subtract_wf,
nat_plus_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
int_seg_properties,
mdist_wf,
nat_plus_inc_int_nzero,
int_nzero-rational,
int-subtype-rationals,
equal_functionality_wrt_subtype_rel2,
int_subtype_base,
istype-int,
less_than_wf,
set_subtype_base,
rationals_wf,
equal-wf-base,
not_functionality_wrt_implies,
rneq-int,
rdiv_wf,
rleq_wf,
int_seg_wf,
equipollent_inversion,
compact-dist-positive,
istype-universe,
metric_wf,
meq_wf,
finite_wf,
mk-metric-space_wf,
mcomplete_wf,
metric-on-subtype,
subtype_rel_self,
mcompact_wf,
mcauchy_wf,
istype-nat,
real_wf,
subtype_rel_dep_function,
Error :stable-union_wf,
mconverges-to_wf,
istype-void,
not_wf,
exists_wf,
all_wf,
or_wf,
double-negation-hyp-elim,
compact-dist_wf,
int-to-real_wf,
rless_wf,
Error :not-not-finite-all-or-exists,
nat_wf
Rules used in proof :
imageMemberEquality,
inrFormation_alt,
cumulativity,
inlFormation_alt,
applyLambdaEquality,
multiplyEquality,
minusEquality,
addEquality,
independent_pairFormation,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
equalityIstype,
imageElimination,
baseClosed,
intEquality,
promote_hyp,
voidElimination,
universeEquality,
instantiate,
functionEquality,
productIsType,
functionIsType,
unionIsType,
unionElimination,
productEquality,
universeIsType,
setIsType,
independent_isectElimination,
because_Cache,
setEquality,
natural_numberEquality,
closedConclusion,
isectElimination,
dependent_set_memberEquality_alt,
dependent_pairFormation_alt,
productElimination,
independent_functionElimination,
extract_by_obid,
introduction,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
rename,
setElimination,
lambdaEquality_alt,
hypothesisEquality,
applyEquality,
functionExtensionality,
thin,
dependent_functionElimination,
hypothesis,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} X {}\mrightarrow{} \mBbbP{}].
finite(T)
{}\mRightarrow{} mcomplete(X with d)
{}\mRightarrow{} (\mforall{}i:T. mcompact(\{x:X| P[i;x]\} ;d))
{}\mRightarrow{} mcomplete(stable-union(X;T;i,x.P[i;x]) with d)
supposing \mforall{}i:T. \mforall{}x,y:X. (P[i;x] {}\mRightarrow{} y \mequiv{} x {}\mRightarrow{} P[i;y])
Date html generated:
2019_10_30-AM-07_13_50
Last ObjectModification:
2019_10_25-PM-06_54_56
Theory : reals
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