Nuprl Lemma : bag-member-filter2
∀[T:Type]. ∀[x:T]. ∀[bs:bag(T)]. ∀[P:{x:T| x ↓∈ bs} ⟶ 𝔹]. uiff(x ↓∈ [x∈bs|P[x]];x ↓∈ bs ∧ (↑P[x]))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-filter: [x∈b|p[x]],
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
bag-member: x ↓∈ bs,
squash: ↓T,
implies: P ⇒ Q
Lemmas referenced :
assert_witness,
bag_wf,
bool_wf,
bag-member-filter-implies1,
assert_wf,
subtype_rel_bag,
bag-member_wf,
bag-filter-wf2,
bag-member-filter-implies2
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
dependent_set_memberEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
productElimination,
independent_pairFormation,
productEquality,
functionEquality,
universeEquality,
isect_memberFormation,
introduction,
independent_pairEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination,
isect_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[bs:bag(T)]. \mforall{}[P:\{x:T| x \mdownarrow{}\mmember{} bs\} {}\mrightarrow{} \mBbbB{}]. uiff(x \mdownarrow{}\mmember{} [x\mmember{}bs|P[x]];x \mdownarrow{}\mmember{} bs \mwedge{} (\muparrow{}P[x])\000C)
Date html generated:
2016_05_15-PM-02_47_51
Last ObjectModification:
2016_01_16-AM-08_43_19
Theory : bags
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