Nuprl Lemma : rv-disjoint-rv-shift
∀p:FinProbSpace. ∀n:ℕ. ∀X,Y:RandomVariable(p;n).
(rv-disjoint(p;n;X;Y)
⇒ (∀x,y:Outcome. (rv-shift(x;X) = rv-shift(y;X) ∈ RandomVariable(p;n - 1)))
∨ (∀x,y:Outcome. (rv-shift(x;Y) = rv-shift(y;Y) ∈ RandomVariable(p;n - 1)))
supposing 0 < n)
Proof
Definitions occuring in Statement :
rv-disjoint: rv-disjoint(p;n;X;Y),
rv-shift: rv-shift(x;X),
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
rv-disjoint: rv-disjoint(p;n;X;Y),
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
or: P ∨ Q,
rv-shift: rv-shift(x;X),
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
subtype_rel: A ⊆r B,
cons-seq: cons-seq(x;s),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
finite-prob-space: FinProbSpace,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s]
Lemmas referenced :
member-less_than,
false_wf,
lelt_wf,
cons-seq_wf,
int_seg_wf,
subtract_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_properties,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
sq_stable__and,
sq_stable__le,
sq_stable__less_than,
length_wf,
rationals_wf,
squash_wf,
less_than_wf,
decidable__le,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
not_wf,
equal-wf-T-base,
p-outcome_wf,
all_wf,
random-variable_wf,
le_wf,
rv-shift_wf,
rv-disjoint_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
sqequalRule,
unionElimination,
inlFormation,
functionExtensionality,
because_Cache,
applyEquality,
independent_functionElimination,
equalityElimination,
productElimination,
voidElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageMemberEquality,
baseClosed,
imageElimination,
functionEquality,
inrFormation
Latex:
\mforall{}p:FinProbSpace. \mforall{}n:\mBbbN{}. \mforall{}X,Y:RandomVariable(p;n).
(rv-disjoint(p;n;X;Y)
{}\mRightarrow{} (\mforall{}x,y:Outcome. (rv-shift(x;X) = rv-shift(y;X)))
\mvee{} (\mforall{}x,y:Outcome. (rv-shift(x;Y) = rv-shift(y;Y)))
supposing 0 < n)
Date html generated:
2018_05_22-AM-00_35_05
Last ObjectModification:
2017_07_26-PM-07_00_01
Theory : randomness
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