Nuprl Lemma : mcompact-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)) ⇒ T ⇒ mcompact(stable-union(X;T;i,x.P[i;x\000C]);d)
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 :
true: True,
surject: Surj(A;B;f),
biject: Bij(A;B;f),
sq_exists: ∃x:A [B[x]],
rless: x < y,
less_than': less_than'(a;b),
equipollent: A ~ B,
finite: finite(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
pi1: fst(t),
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
stable-union: Error :stable-union,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
mcompact: mcompact(X;d),
implies: P ⇒ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rless_irreflexivity,
rless_transitivity1,
real_wf,
iff_weakening_equal,
subtype_rel_sets_simple,
true_wf,
squash_wf,
equal_wf,
rleq_weakening_rless,
int_seg-case,
rless-int-fractions,
rless-cases,
istype-less_than,
nat_plus_properties,
istype-false,
equipollent_inversion,
nat_plus_subtype_nat,
nsub_finite,
finite-product,
rless_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_seg_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
mdist_wf,
rleq_wf,
nat_plus_wf,
int_seg_wf,
istype-universe,
metric_wf,
meq_wf,
subtype_rel_self,
finite_wf,
mk-metric-space_wf,
mcomplete_wf,
mcompact_wf,
istype-nat,
istype-le,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
metric-on-subtype,
Error :stable-union_wf,
m-TB-iff,
mcomplete-stable-union
Rules used in proof :
baseClosed,
imageMemberEquality,
hyp_replacement,
spreadEquality,
inlFormation_alt,
applyLambdaEquality,
dependent_pairEquality_alt,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
imageElimination,
inrFormation_alt,
closedConclusion,
productIsType,
functionExtensionality,
universeEquality,
instantiate,
functionIsType,
promote_hyp,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
addEquality,
natural_numberEquality,
multiplyEquality,
dependent_set_memberEquality_alt,
setIsType,
setEquality,
setElimination,
inhabitedIsType,
universeIsType,
applyEquality,
lambdaEquality_alt,
sqequalRule,
because_Cache,
productElimination,
dependent_functionElimination,
independent_pairFormation,
rename,
independent_functionElimination,
independent_isectElimination,
lambdaFormation_alt,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
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{} T
{}\mRightarrow{} mcompact(stable-union(X;T;i,x.P[i;x]);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_14_15
Last ObjectModification:
2019_10_25-PM-09_57_41
Theory : reals
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