Nuprl Lemma : member-p-union
∀p:FinProbSpace. ∀A,B:p-open(p). ∀s:ℕ ⟶ Outcome. (s ∈ p-union(A;B) ⇐⇒ s ∈ A ∨ s ∈ B)
Proof
Definitions occuring in Statement :
p-union: p-union(A;B),
p-open-member: s ∈ C,
p-open: p-open(p),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
p-open-member: s ∈ C,
p-union: p-union(A;B),
p-open: p-open(p),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
int_seg: {i..j-},
rev_implies: P ⇐ Q,
eq_int: (i =z j)
Lemmas referenced :
exists_wf,
nat_wf,
equal-wf-T-base,
eq_int_wf,
subtype_rel_dep_function,
p-outcome_wf,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
or_wf,
set_wf,
all_wf,
le_wf,
finite-prob-space_wf,
assert_wf,
bnot_wf,
not_wf,
bfalse_wf,
assert_elim,
and_wf,
btrue_neq_bfalse,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
int_subtype_base,
uiff_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
lambdaEquality,
because_Cache,
applyEquality,
setElimination,
rename,
hypothesisEquality,
dependent_pairEquality,
natural_numberEquality,
independent_isectElimination,
functionEquality,
unionElimination,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
baseClosed,
productEquality,
functionExtensionality,
addLevel,
levelHypothesis,
dependent_set_memberEquality,
applyLambdaEquality,
impliesFunctionality,
inlFormation,
inrFormation,
intEquality
Latex:
\mforall{}p:FinProbSpace. \mforall{}A,B:p-open(p). \mforall{}s:\mBbbN{} {}\mrightarrow{} Outcome. (s \mmember{} p-union(A;B) \mLeftarrow{}{}\mRightarrow{} s \mmember{} A \mvee{} s \mmember{} B)
Date html generated:
2018_05_22-AM-00_36_37
Last ObjectModification:
2017_07_26-PM-07_00_31
Theory : randomness
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