Nuprl Lemma : member-countable-p-union
∀p:FinProbSpace. ∀A:ℕ ⟶ p-open(p). ∀s:ℕ ⟶ Outcome. ((∃i:ℕ. s ∈ A[i]) ⇒ s ∈ countable-p-union(i.A[i]))
Proof
Definitions occuring in Statement :
countable-p-union: countable-p-union(i.A[i]),
p-open-member: s ∈ C,
p-open: p-open(p),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
p-open-member: s ∈ C,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
countable-p-union: countable-p-union(i.A[i]),
pi1: fst(t),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
p-open: p-open(p)
Lemmas referenced :
exists_wf,
nat_wf,
p-open-member_wf,
p-outcome_wf,
p-open_wf,
finite-prob-space_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
intformless_wf,
int_formula_prop_less_lemma,
decidable__lt,
equal-wf-T-base,
subtype_rel_dep_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
eq_int_wf,
assert_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
not_wf,
uiff_transitivity,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
imax-list-ub,
map_wf,
upto_wf,
map-length,
length_upto,
l_exists_iff,
l_member_wf,
lelt_wf,
member_map,
equal-wf-base-T,
member_upto2,
decidable__equal_int,
int_seg_properties,
imax-list-lb,
l_all_iff,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
functionEquality,
dependent_pairFormation,
setElimination,
rename,
because_Cache,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
dependent_functionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
productEquality,
dependent_pairEquality,
baseClosed,
impliesFunctionality,
setEquality,
imageMemberEquality,
applyLambdaEquality,
addLevel,
allFunctionality
Latex:
\mforall{}p:FinProbSpace. \mforall{}A:\mBbbN{} {}\mrightarrow{} p-open(p). \mforall{}s:\mBbbN{} {}\mrightarrow{} Outcome.
((\mexists{}i:\mBbbN{}. s \mmember{} A[i]) {}\mRightarrow{} s \mmember{} countable-p-union(i.A[i]))
Date html generated:
2018_05_22-AM-00_36_57
Last ObjectModification:
2017_07_26-PM-07_00_33
Theory : randomness
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