Nuprl Lemma : expectation-qsum
∀[k:ℕ]. ∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[X:ℕk ⟶ RandomVariable(p;n)].
(E(n;λs.Σ0 ≤ i < k. X i s) = Σ0 ≤ i < k. E(n;X i) ∈ ℚ)
Proof
Definitions occuring in Statement :
expectation: E(n;F),
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
qsum: Σa ≤ j < b. E[j],
rationals: ℚ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
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,
decidable: Dec(P),
or: P ∨ Q,
qsum: Σa ≤ j < b. E[j],
rng_sum: rng_sum,
mon_itop: Π lb ≤ i < ub. E[i],
add_grp_of_rng: r↓+gp,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
qrng: <ℚ+*>,
rng_plus: +r,
rng_zero: 0,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
le: A ≤ B,
random-variable: RandomVariable(p;n),
so_lambda: λ2x.t[x],
so_apply: x[s],
finite-prob-space: FinProbSpace,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
infix_ap: x f y,
rv-add: X + Y,
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
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,
int_seg_wf,
random-variable_wf,
nat_wf,
finite-prob-space_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
lt_int_wf,
bool_wf,
equal-wf-base,
assert_wf,
le_int_wf,
le_wf,
bnot_wf,
expectation-constant,
int-subtype-rationals,
top_wf,
subtype_rel_dep_function,
length_wf,
rationals_wf,
p-outcome_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
int_subtype_base,
squash_wf,
true_wf,
expectation-rv-add,
qsum_wf,
decidable__lt,
lelt_wf,
qadd_wf,
expectation_wf,
iff_weakening_equal,
int_seg_subtype,
false_wf,
subtype_rel_self
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,
functionEquality,
because_Cache,
unionElimination,
baseClosed,
imageElimination,
productElimination,
applyEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
universeEquality,
functionExtensionality,
dependent_set_memberEquality,
imageMemberEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[p:FinProbSpace]. \mforall{}[n:\mBbbN{}]. \mforall{}[X:\mBbbN{}k {}\mrightarrow{} RandomVariable(p;n)].
(E(n;\mlambda{}s.\mSigma{}0 \mleq{} i < k. X i s) = \mSigma{}0 \mleq{} i < k. E(n;X i))
Date html generated:
2018_05_22-AM-00_34_55
Last ObjectModification:
2017_07_26-PM-06_59_57
Theory : randomness
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