Nuprl Lemma : rv-disjoint-shift
∀p:FinProbSpace. ∀n:ℕ. ∀X,Y:RandomVariable(p;n). ∀x:Outcome.
rv-disjoint(p;n;X;Y) ⇒ rv-disjoint(p;n - 1;rv-shift(x;X);rv-shift(x;Y)) 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,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
rv-disjoint: rv-disjoint(p;n;X;Y),
int_seg: {i..j-},
uiff: uiff(P;Q),
and: P ∧ Q,
lelt: i ≤ j < k,
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
prop: ℙ,
sq_stable: SqStable(P),
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
subtract: n - m,
le: A ≤ B,
less_than: a < b,
rv-shift: rv-shift(x;X),
guard: {T},
subtype_rel: A ⊆r B,
cons-seq: cons-seq(x;s),
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
random-variable: RandomVariable(p;n),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
member-less_than,
add-member-int_seg2,
sq_stable__and,
le_wf,
less_than_wf,
length_wf,
rationals_wf,
sq_stable__le,
sq_stable__less_than,
squash_wf,
nat_properties,
decidable__le,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
cons-seq_wf,
p-outcome_wf,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
subtract-add-cancel,
subtype_rel_self,
int_seg_wf,
not_wf,
equal_wf,
all_wf,
rv-shift_wf,
random-variable_wf,
rv-disjoint_wf,
nat_wf,
finite-prob-space_wf,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
true_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
iff_weakening_equal,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
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,
because_Cache,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
isect_memberEquality,
independent_functionElimination,
sqequalRule,
lambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
inlFormation,
functionExtensionality,
applyEquality,
functionEquality,
addEquality,
inrFormation,
equalityTransitivity,
equalitySymmetry,
universeEquality,
equalityElimination,
impliesFunctionality
Latex:
\mforall{}p:FinProbSpace. \mforall{}n:\mBbbN{}. \mforall{}X,Y:RandomVariable(p;n). \mforall{}x:Outcome.
rv-disjoint(p;n;X;Y) {}\mRightarrow{} rv-disjoint(p;n - 1;rv-shift(x;X);rv-shift(x;Y)) supposing 0 < n
Date html generated:
2018_05_22-AM-00_35_26
Last ObjectModification:
2017_07_26-PM-07_00_12
Theory : randomness
Home
Index