Nuprl Lemma : rv-disjoint-compose
∀F,G:ℚ ⟶ ℚ. ∀p:FinProbSpace. ∀n:ℕ. ∀X,Y:RandomVariable(p;n).
(rv-disjoint(p;n;X;Y) ⇒ rv-disjoint(p;n;(X.F[X]) o X;(Y.G[Y]) o Y))
Proof
Definitions occuring in Statement :
rv-compose: (x.F[x]) o X,
rv-disjoint: rv-disjoint(p;n;X;Y),
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
rationals: ℚ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-disjoint: rv-disjoint(p;n;X;Y),
member: t ∈ T,
or: P ∨ Q,
rv-compose: (x.F[x]) o X,
so_apply: x[s],
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
guard: {T}
Lemmas referenced :
all_wf,
int_seg_wf,
not_wf,
equal_wf,
p-outcome_wf,
rationals_wf,
rv-compose_wf,
random-variable_wf,
rv-disjoint_wf,
nat_wf,
finite-prob-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
unionElimination,
inlFormation,
independent_functionElimination,
sqequalRule,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
lambdaEquality,
functionEquality,
intEquality,
functionExtensionality,
inrFormation
Latex:
\mforall{}F,G:\mBbbQ{} {}\mrightarrow{} \mBbbQ{}. \mforall{}p:FinProbSpace. \mforall{}n:\mBbbN{}. \mforall{}X,Y:RandomVariable(p;n).
(rv-disjoint(p;n;X;Y) {}\mRightarrow{} rv-disjoint(p;n;(X.F[X]) o X;(Y.G[Y]) o Y))
Date html generated:
2018_05_22-AM-00_35_22
Last ObjectModification:
2017_07_26-PM-07_00_09
Theory : randomness
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