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
Lemmas :
member-less_than,
false_wf,
lelt_wf,
cons-seq_wf,
trivial-int-eq1,
subtype_rel_self,
int_seg_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,
subtract_wf,
decidable__le,
not-le-2,
not-equal-2,
sq_stable__le,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
minus-one-mul,
minus-minus,
add-swap,
decidable__lt,
less-iff-le,
le-add-cancel-alt,
not_wf,
equal-wf-T-base,
all_wf,
p-outcome_wf,
random-variable_wf,
le_wf,
rv-shift_wf,
less_than_wf,
rv-disjoint_wf,
nat_wf
\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:
2015_07_17-AM-07_59_06
Last ObjectModification:
2015_01_27-AM-11_23_15
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