Nuprl Lemma : rv-iid-add
∀p:FinProbSpace. ∀f:ℕ ⟶ ℕ. ∀X,Y:n:ℕ ⟶ RandomVariable(p;f[n]).
(rv-iid(p;n.f[n];n.X[n])
⇒ rv-iid(p;n.f[n];n.Y[n])
⇒ (∀n:ℕ. ∀i:ℕn + 1. (rv-disjoint(p;f[n];Y[i];X[n]) ∧ rv-disjoint(p;f[n];X[i];Y[n])))
⇒ rv-iid(p;n.f[n];n.X[n] + Y[n]))
Proof
Definitions occuring in Statement :
rv-iid: rv-iid(p;n.f[n];i.X[i]),
rv-disjoint: rv-disjoint(p;n;X;Y),
rv-add: X + Y,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
rv-iid: rv-iid(p;n.f[n];i.X[i]),
so_lambda: λ2x.t[x],
rv-identically-distributed: rv-identically-distributed(p;n.f[n];i.X[i]),
squash: ↓T,
true: True,
rv-compose: (x.F[x]) o X,
rv-mul: X * Y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
less_than: a < b
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
nat_properties,
decidable__le,
istype-le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_weakening2,
subtype_rel-random-variable,
int_seg_subtype_nat,
istype-false,
int_seg_wf,
istype-nat,
rv-disjoint_wf,
rv-iid_wf,
random-variable_wf,
finite-prob-space_wf,
decidable__lt,
intformand_wf,
intformless_wf,
intformeq_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
istype-less_than,
qadd_wf,
squash_wf,
true_wf,
rationals_wf,
qmul_wf,
int-subtype-rationals,
rv-mul_wf,
rv-disjoint-rv-mul2,
rv-disjoint-rv-mul,
equal_wf,
istype-universe,
expectation-rv-add,
subtype_rel_self,
iff_weakening_equal,
expectation-rv-add-squared,
expectation-rv-add-cubed,
expectation-rv-add-fourth,
expectation-rv-disjoint,
rv-add_wf,
rv-disjoint-rv-add2,
rv-disjoint-rv-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
unionElimination,
promote_hyp,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
addEquality,
productElimination,
applyEquality,
dependent_set_memberEquality_alt,
sqequalRule,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
independent_pairFormation,
functionIsType,
productIsType,
inhabitedIsType,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}p:FinProbSpace. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}X,Y:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n]).
(rv-iid(p;n.f[n];n.X[n])
{}\mRightarrow{} rv-iid(p;n.f[n];n.Y[n])
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n + 1. (rv-disjoint(p;f[n];Y[i];X[n]) \mwedge{} rv-disjoint(p;f[n];X[i];Y[n])))
{}\mRightarrow{} rv-iid(p;n.f[n];n.X[n] + Y[n]))
Date html generated:
2019_10_16-PM-00_39_18
Last ObjectModification:
2018_10_18-PM-06_02_59
Theory : randomness
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