Nuprl Lemma : countable-p-union_wf
∀[p:FinProbSpace]. ∀[A:ℕ ⟶ p-open(p)]. (countable-p-union(i.A[i]) ∈ p-open(p))
Proof
Definitions occuring in Statement :
countable-p-union: countable-p-union(i.A[i]),
p-open: p-open(p),
finite-prob-space: FinProbSpace,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
countable-p-union: countable-p-union(i.A[i]),
p-open: p-open(p),
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
so_apply: x[s],
int_seg: {i..j-},
all: ∀x:A. B[x],
pi1: fst(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
cand: A c∧ B,
guard: {T},
sq_type: SQType(T),
sq_stable: SqStable(P)
Lemmas referenced :
nat_wf,
int_seg_wf,
p-outcome_wf,
all_wf,
le_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_dep_function,
subtype_rel_self,
finite-prob-space_wf,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
lelt_wf,
imax-list_wf,
map_wf,
upto_wf,
map-length,
length_upto,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
imax-list-ub,
l_exists_iff,
l_member_wf,
decidable__le,
int_seg_properties,
member_map,
subtype_rel_list,
member_upto,
imax-list-lb,
l_all_iff,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
member_upto2,
set_subtype_base,
set_wf,
sq_stable__le
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
comment,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
extract_by_obid,
setEquality,
productEquality,
because_Cache,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
dependent_pairEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
isect_memberEquality,
productElimination,
baseClosed,
intEquality,
unionElimination,
equalityElimination,
independent_functionElimination,
impliesFunctionality,
dependent_functionElimination,
imageMemberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
applyLambdaEquality,
instantiate,
cumulativity,
promote_hyp,
imageElimination
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[A:\mBbbN{} {}\mrightarrow{} p-open(p)]. (countable-p-union(i.A[i]) \mmember{} p-open(p))
Date html generated:
2018_05_22-AM-00_36_50
Last ObjectModification:
2017_07_26-PM-07_00_32
Theory : randomness
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