Nuprl Lemma : rv-disjoint-monotone-in-first
∀p:FinProbSpace. ∀n:ℕ. ∀X,Y:RandomVariable(p;n). (rv-disjoint(p;n;X;Y) ⇒ (∀m:ℕ. rv-disjoint(p;m;X;Y) supposing n ≤ m))
Proof
Definitions occuring in Statement :
rv-disjoint: rv-disjoint(p;n;X;Y),
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
rv-disjoint: rv-disjoint(p;n;X;Y),
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
p-outcome: Outcome,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
less_than'_wf,
decidable__lt,
int_seg_wf,
le_wf,
nat_wf,
rv-disjoint_wf,
random-variable_wf,
finite-prob-space_wf,
lelt_wf,
subtype_rel_function,
p-outcome_wf,
int_seg_subtype,
false_wf,
subtype_rel_self,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
all_wf,
not_wf,
equal_wf,
rationals_wf,
subtype_rel_dep_function,
length_wf,
squash_wf,
true_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
natural_numberEquality,
dependent_set_memberEquality,
independent_pairFormation,
inlFormation,
applyEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
functionEquality,
inrFormation,
functionExtensionality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
instantiate
Latex:
\mforall{}p:FinProbSpace. \mforall{}n:\mBbbN{}. \mforall{}X,Y:RandomVariable(p;n).
(rv-disjoint(p;n;X;Y) {}\mRightarrow{} (\mforall{}m:\mBbbN{}. rv-disjoint(p;m;X;Y) supposing n \mleq{} m))
Date html generated:
2018_05_22-AM-00_35_20
Last ObjectModification:
2018_05_19-PM-03_56_24
Theory : randomness
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