Nuprl Lemma : rv-iid-add-const
∀p:FinProbSpace. ∀a:ℚ. ∀f:ℕ ⟶ ℕ. ∀X:n:ℕ ⟶ RandomVariable(p;f[n]).
(rv-iid(p;n.f[n];n.X[n]) ⇒ rv-iid(p;n.f[n];n.X[n] + a))
Proof
Definitions occuring in Statement :
rv-iid: rv-iid(p;n.f[n];i.X[i]),
rv-const: a,
rv-add: X + Y,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
rationals: ℚ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uall: ∀[x:A]. B[x],
and: P ∧ Q,
cand: A c∧ B,
nat: ℕ,
prop: ℙ,
rv-iid: rv-iid(p;n.f[n];i.X[i]),
guard: {T},
rv-identically-distributed: rv-identically-distributed(p;n.f[n];i.X[i]),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rv-const: a,
rv-compose: (x.F[x]) o X,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b
Lemmas referenced :
rv-iid-add,
nat_wf,
rv-const_wf,
rv-disjoint-const,
int_seg_wf,
rv-iid_wf,
random-variable_wf,
rationals_wf,
finite-prob-space_wf,
rv-disjoint-symmetry,
equal_wf,
squash_wf,
true_wf,
expectation-rv-const,
expectation_wf,
iff_weakening_equal,
qmul_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel-random-variable,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
nat_properties,
decidable__le,
le_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_weakening2,
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,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
isectElimination,
independent_functionElimination,
independent_pairFormation,
natural_numberEquality,
addEquality,
setElimination,
rename,
functionEquality,
because_Cache,
productElimination,
promote_hyp,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
unionElimination,
instantiate,
cumulativity,
intEquality,
dependent_set_memberEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}p:FinProbSpace. \mforall{}a:\mBbbQ{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}X:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n]).
(rv-iid(p;n.f[n];n.X[n]) {}\mRightarrow{} rv-iid(p;n.f[n];n.X[n] + a))
Date html generated:
2018_05_22-AM-00_38_01
Last ObjectModification:
2017_07_26-PM-07_00_45
Theory : randomness
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