Nuprl Lemma : rv-shift-linear
∀[k:FinProbSpace]. ∀[x:Outcome]. ∀[n:ℕ+]. ∀[X,Y:RandomVariable(k;n)]. ∀[a,b:ℚ].
(rv-shift(x;a*X + b*Y) = a*rv-shift(x;X) + b*rv-shift(x;Y) ∈ RandomVariable(k;n - 1))
Proof
Definitions occuring in Statement :
rv-shift: rv-shift(x;X),
rv-scale: q*X,
rv-add: X + Y,
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
rv-shift: rv-shift(x;X),
rv-scale: q*X,
rv-add: X + Y,
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
nat_plus: ℕ+,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
less_than: a < b,
squash: ↓T,
so_apply: x[s]
Lemmas referenced :
int-subtype-rationals,
qle_wf,
l_member_wf,
l_all_wf2,
decidable__lt,
select_wf,
qsum_wf,
equal-wf-T-base,
list_wf,
set_wf,
nat_plus_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
nat_plus_properties,
subtract_wf,
rationals_wf,
length_wf,
int_seg_wf,
cons-seq_wf,
qmul_wf,
qadd_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
because_Cache,
functionEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
imageElimination,
baseClosed,
lambdaFormation,
setEquality
Latex:
\mforall{}[k:FinProbSpace]. \mforall{}[x:Outcome]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[X,Y:RandomVariable(k;n)]. \mforall{}[a,b:\mBbbQ{}].
(rv-shift(x;a*X + b*Y) = a*rv-shift(x;X) + b*rv-shift(x;Y))
Date html generated:
2016_05_15-PM-11_46_20
Last ObjectModification:
2016_01_17-AM-10_07_05
Theory : randomness
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