Nuprl Lemma : expectation-imax-list
∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[k:ℕ+]. ∀[X:ℕk ⟶ (ℕn ⟶ Outcome) ⟶ ℕ].
(E(n;λs.imax-list(map(λi.(X i s);upto(k)))) ≤ Σ0 ≤ i < k. E(n;X i))
Proof
Definitions occuring in Statement :
expectation: E(n;F),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
upto: upto(n),
imax-list: imax-list(L),
map: map(f;as),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
and: P ∧ Q,
prop: ℙ,
random-variable: RandomVariable(p;n),
so_lambda: λ2x.t[x],
so_apply: x[s],
finite-prob-space: FinProbSpace,
p-outcome: Outcome,
rv-le: X ≤ Y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
squash: ↓T,
true: True
Lemmas referenced :
qle_witness,
expectation_wf,
imax-list_wf,
map_wf,
int_seg_wf,
upto_wf,
map-length,
length_upto,
nat_plus_subtype_nat,
nat_plus_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
subtype_rel_dep_function,
p-outcome_wf,
length_wf,
rationals_wf,
int-subtype-rationals,
qsum_wf,
nat_wf,
subtype_rel_set,
le_wf,
nat_plus_wf,
finite-prob-space_wf,
expectation-monotone,
length-map,
qsum-int,
qle-int,
imax-list-lb,
l_all_iff,
l_member_wf,
exists_wf,
equal_wf,
member_map,
all_wf,
summand-qle-sum,
non_neg_length,
map_length,
int_seg_properties,
decidable__le,
lelt_wf,
length_wf_nat,
intformle_wf,
int_formula_prop_le_lemma,
qle_wf,
squash_wf,
true_wf,
expectation-qsum,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
intEquality,
applyEquality,
functionExtensionality,
sqequalRule,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
functionEquality,
lambdaFormation,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
setEquality,
productEquality,
addLevel,
allFunctionality,
impliesFunctionality,
applyLambdaEquality,
dependent_set_memberEquality,
hyp_replacement,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[n:\mBbbN{}]. \mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[X:\mBbbN{}k {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} Outcome) {}\mrightarrow{} \mBbbN{}].
(E(n;\mlambda{}s.imax-list(map(\mlambda{}i.(X i s);upto(k)))) \mleq{} \mSigma{}0 \mleq{} i < k. E(n;X i))
Date html generated:
2018_05_22-AM-00_36_04
Last ObjectModification:
2017_07_26-PM-07_00_23
Theory : randomness
Home
Index