Nuprl Lemma : nullset-monotone
∀p:FinProbSpace. ∀[P,Q:(ℕ ⟶ Outcome) ⟶ ℙ]. ((∀s:ℕ ⟶ Outcome. (Q[s] ⇒ P[s])) ⇒ nullset(p;P) ⇒ nullset(p;Q))
Proof
Definitions occuring in Statement :
nullset: nullset(p;S),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
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],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
nullset: nullset(p;S),
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
so_apply: x[s],
prop: ℙ,
cand: A c∧ B,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B
Lemmas referenced :
nat_wf,
p-outcome_wf,
all_wf,
p-open-member_wf,
p-measure-le_wf,
rationals_wf,
qless_wf,
int-subtype-rationals,
nullset_wf,
finite-prob-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
dependent_pairFormation,
independent_pairFormation,
promote_hyp,
independent_functionElimination,
because_Cache,
applyEquality,
functionEquality,
lemma_by_obid,
isectElimination,
productEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
natural_numberEquality,
cumulativity,
universeEquality
Latex:
\mforall{}p:FinProbSpace
\mforall{}[P,Q:(\mBbbN{} {}\mrightarrow{} Outcome) {}\mrightarrow{} \mBbbP{}]. ((\mforall{}s:\mBbbN{} {}\mrightarrow{} Outcome. (Q[s] {}\mRightarrow{} P[s])) {}\mRightarrow{} nullset(p;P) {}\mRightarrow{} nullset(p;Q))
Date html generated:
2016_05_15-PM-11_50_35
Last ObjectModification:
2015_12_28-PM-07_14_34
Theory : randomness
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