Nuprl Lemma : expectation-rv-disjoint
∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[X,Y:RandomVariable(p;n)].
  E(n;X * Y) = (E(n;X) * E(n;Y)) ∈ ℚ supposing rv-disjoint(p;n;X;Y)
Proof
Definitions occuring in Statement : 
rv-disjoint: rv-disjoint(p;n;X;Y)
, 
expectation: E(n;F)
, 
rv-mul: X * Y
, 
random-variable: RandomVariable(p;n)
, 
finite-prob-space: FinProbSpace
, 
qmul: r * s
, 
rationals: ℚ
, 
nat: ℕ
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
nat: ℕ
, 
implies: P 
⇒ Q
, 
false: False
, 
ge: i ≥ j 
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
all: ∀x:A. B[x]
, 
top: Top
, 
and: P ∧ Q
, 
prop: ℙ
, 
expectation: E(n;F)
, 
ycomb: Y
, 
eq_int: (i =z j)
, 
subtract: n - m
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
subtype_rel: A ⊆r B
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
uiff: uiff(P;Q)
, 
bfalse: ff
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
rv-mul: X * Y
, 
random-variable: RandomVariable(p;n)
, 
finite-prob-space: FinProbSpace
, 
p-outcome: Outcome
, 
int_seg: {i..j-}
, 
sq_stable: SqStable(P)
, 
lelt: i ≤ j < k
, 
squash: ↓T
, 
nat_plus: ℕ+
, 
true: True
, 
guard: {T}
, 
qmul: r * s
, 
callbyvalueall: callbyvalueall, 
evalall: evalall(t)
, 
rv-shift: rv-shift(x;X)
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
sq_type: SQType(T)
Lemmas referenced : 
nat_properties, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformless_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_wf, 
ge_wf, 
less_than_wf, 
rv-disjoint_wf, 
random-variable_wf, 
false_wf, 
le_wf, 
decidable__le, 
subtract_wf, 
intformnot_wf, 
itermSubtract_wf, 
int_formula_prop_not_lemma, 
int_term_value_subtract_lemma, 
nat_wf, 
finite-prob-space_wf, 
eq_int_wf, 
bool_wf, 
equal-wf-base, 
int_subtype_base, 
assert_wf, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
bnot_wf, 
not_wf, 
uiff_transitivity, 
eqtt_to_assert, 
assert_of_eq_int, 
iff_transitivity, 
iff_weakening_uiff, 
eqff_to_assert, 
assert_of_bnot, 
equal_wf, 
qmul_wf, 
null-seq_wf, 
int_seg_wf, 
length_wf, 
rationals_wf, 
rv-disjoint-rv-shift, 
weighted-sum_wf2, 
expectation_wf, 
sq_stable__and, 
sq_stable__le, 
sq_stable__less_than, 
member-less_than, 
squash_wf, 
rv-shift_wf, 
rv-mul_wf, 
p-outcome_wf, 
ws-constant, 
natural_number_wf_p-outcome, 
true_wf, 
rv-disjoint-shift, 
decidable__equal_int, 
cons-seq_wf, 
subtype_rel_dep_function, 
subtype_base_sq, 
add-associates, 
add-swap, 
add-commutes, 
zero-add, 
int-subtype-rationals, 
qadd_comm_q, 
mon_ident_q, 
iff_weakening_equal, 
ws-linear, 
qadd_wf, 
qmul_zero_qrng, 
qmul_com, 
qmul_comm_qrng
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
setElimination, 
rename, 
intWeakElimination, 
lambdaFormation, 
natural_numberEquality, 
independent_isectElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
dependent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
sqequalRule, 
independent_pairFormation, 
computeAll, 
independent_functionElimination, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
dependent_set_memberEquality, 
because_Cache, 
unionElimination, 
baseApply, 
closedConclusion, 
baseClosed, 
applyEquality, 
equalityElimination, 
productElimination, 
impliesFunctionality, 
hyp_replacement, 
applyLambdaEquality, 
imageMemberEquality, 
imageElimination, 
functionExtensionality, 
universeEquality, 
addEquality, 
instantiate, 
cumulativity, 
functionEquality
Latex:
\mforall{}[p:FinProbSpace].  \mforall{}[n:\mBbbN{}].  \mforall{}[X,Y:RandomVariable(p;n)].
    E(n;X  *  Y)  =  (E(n;X)  *  E(n;Y))  supposing  rv-disjoint(p;n;X;Y)
Date html generated:
2018_05_22-AM-00_35_41
Last ObjectModification:
2017_07_26-PM-07_00_14
Theory : randomness
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