Nuprl Lemma : nullset-union
∀p:FinProbSpace. ∀[S:ℕ ⟶ (ℕ ⟶ Outcome) ⟶ ℙ]. ((∀i:ℕ. nullset(p;S[i])) ⇒ nullset(p;λs.∃i:ℕ. (S[i] s)))
Proof
Definitions occuring in Statement :
nullset: nullset(p;S),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
nullset: nullset(p;S),
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
top: Top,
cand: A c∧ B,
pi1: fst(t),
guard: {T},
uimplies: b supposing a,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
false: False,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
p-measure-le: measure(C) ≤ q,
random-variable: RandomVariable(p;n),
p-open: p-open(p),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
countable-p-union: countable-p-union(i.A[i]),
qdiv: (r/s),
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
qinv: 1/r,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
nat_plus: ℕ+,
le: A ≤ B,
subtract: n - m,
int_seg: {i..j-},
lelt: i ≤ j < k,
qeq: qeq(r;s),
eq_int: (i =z j),
qsub: r - s,
qadd: r + s,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
qless_wf,
rationals_wf,
nat_wf,
all_wf,
exists_wf,
p-open-member_wf,
p-measure-le_wf,
p-open_wf,
p-outcome_wf,
pi1_wf_top,
equal_wf,
set_wf,
int-subtype-rationals,
nullset_wf,
finite-prob-space_wf,
countable-p-union_wf,
qmul_wf,
qexp_wf,
qdiv_wf,
int_nzero-rational,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
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,
qmul-positive,
sq_stable_from_decidable,
decidable__qless,
qexp-positive-iff,
qinv-positive,
qless-int,
assert_wf,
isEven_wf,
equal-wf-T-base,
member-countable-p-union,
qless_witness,
expectation_wf,
int_seg_wf,
length_wf,
eq_int_wf,
bool_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,
expectation-constant,
top_wf,
subtype_rel_dep_function,
qexp-positive,
expectation-imax-list,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less_than_wf,
int_seg_subtype_nat,
int_seg_properties,
qsum-qle,
qle_weakening_lt_qorder,
squash_wf,
prod_sum_l_q,
iff_weakening_equal,
qsum_wf,
qmul_com,
qexp_step,
q-geometric-series,
qsub_wf,
assert-qeq,
qadd_wf,
qmul_ac_1_qrng,
qmul_comm_qrng,
qmul-qdiv-cancel3,
qmul_over_plus_qrng,
qmul_over_minus_qrng,
qmul_one_qrng,
qadd_preserves_qless,
qadd_ac_1_q,
qadd_comm_q,
qadd_inv_assoc_q,
qinverse_q,
mon_ident_q,
qless_transitivity_1_qorder,
imax-list_wf,
map_wf,
upto_wf,
map-length,
length_upto,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
rename,
dependent_pairFormation,
lambdaEquality,
introduction,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
applyEquality,
because_Cache,
hypothesisEquality,
setElimination,
functionExtensionality,
productEquality,
functionEquality,
dependent_set_memberEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
setEquality,
independent_pairFormation,
cumulativity,
universeEquality,
independent_isectElimination,
addLevel,
instantiate,
intEquality,
baseClosed,
addEquality,
unionElimination,
int_eqEquality,
computeAll,
inlFormation,
imageMemberEquality,
imageElimination,
inrFormation,
minusEquality,
dependent_pairEquality,
equalityElimination,
impliesFunctionality,
hyp_replacement,
applyLambdaEquality,
promote_hyp
Latex:
\mforall{}p:FinProbSpace
\mforall{}[S:\mBbbN{} {}\mrightarrow{} (\mBbbN{} {}\mrightarrow{} Outcome) {}\mrightarrow{} \mBbbP{}]. ((\mforall{}i:\mBbbN{}. nullset(p;S[i])) {}\mRightarrow{} nullset(p;\mlambda{}s.\mexists{}i:\mBbbN{}. (S[i] s)))
Date html generated:
2018_05_22-AM-00_37_18
Last ObjectModification:
2017_07_26-PM-07_00_37
Theory : randomness
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