Nuprl Lemma : p-union_wf
∀[p:FinProbSpace]. ∀[A,B:p-open(p)]. (p-union(A;B) ∈ p-open(p))
Proof
Definitions occuring in Statement :
p-union: p-union(A;B),
p-open: p-open(p),
finite-prob-space: FinProbSpace,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
p-union: p-union(A;B),
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],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
set_wf,
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,
eqtt_to_assert,
assert_of_eq_int,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
assert_elim,
btrue_neq_bfalse,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
le_weakening2,
decidable__lt,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
functionEquality,
productEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
dependent_pairEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
isect_memberEquality,
dependent_set_memberEquality,
productElimination,
unionElimination,
equalityElimination,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
intEquality,
addLevel,
levelHypothesis,
applyLambdaEquality,
int_eqEquality,
voidEquality,
computeAll,
impliesFunctionality
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[A,B:p-open(p)]. (p-union(A;B) \mmember{} p-open(p))
Date html generated:
2018_05_22-AM-00_36_22
Last ObjectModification:
2017_07_26-PM-07_00_28
Theory : randomness
Home
Index