Nuprl Lemma : bag-split
∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[as:bag(T)]. (as = ([x∈as|p[x]] + [x∈as|¬bp[x]]) ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-filter: [x∈b|p[x]],
bag-append: as + bs,
bag: bag(T),
bnot: ¬bb,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
bag_wf,
bag-filter-split,
iff_weakening_equal,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
sqequalRule,
functionExtensionality,
cumulativity,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[as:bag(T)]. (as = ([x\mmember{}as|p[x]] + [x\mmember{}as|\mneg{}\msubb{}p[x]]))
Date html generated:
2017_10_01-AM-08_45_29
Last ObjectModification:
2017_07_26-PM-04_30_46
Theory : bags
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