### Nuprl Lemma : bag-member-filter

`∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[x:T]. ∀[bs:bag(T)].  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` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` squash: `↓T` sq_stable: `SqStable(P)` implies: `P `` Q` exists: `∃x:A. B[x]` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` bag-filter: `[x∈b|p[x]]` bag-member: `x ↓∈ bs` rev_uimplies: `rev_uimplies(P;Q)` all: `∀x:A. B[x]` bag: `bag(T)` quotient: `x,y:A//B[x; y]` cand: `A c∧ B` guard: `{T}` iff: `P `⇐⇒` Q` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` rev_implies: `P `` Q` label: `...\$L... t`
Lemmas referenced :  bag_to_squash_list sq_stable__bag-member bag-member_wf bag-filter_wf subtype_rel_bag assert_wf assert_witness bag_wf bool_wf eqtt_to_assert sq_stable_from_decidable decidable__assert member-permutation member_filter_2 l_member_wf equal_wf list-subtype-bag member_wf list_wf filter_wf5 permutation_wf member_filter iff_imp_equal_bool true_wf assert_functionality_wrt_uiff
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality imageElimination hypothesis independent_functionElimination productElimination promote_hyp equalitySymmetry hyp_replacement applyLambdaEquality cumulativity sqequalRule lambdaEquality applyEquality functionExtensionality setEquality independent_isectElimination setElimination rename imageMemberEquality baseClosed independent_pairEquality productEquality isect_memberEquality equalityTransitivity functionEquality dependent_functionElimination pertypeElimination dependent_pairFormation lambdaFormation natural_numberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[x:T].  \mforall{}[bs:bag(T)].    uiff(x  \mdownarrow{}\mmember{}  [x\mmember{}bs|P[x]];x  \mdownarrow{}\mmember{}  bs  \mwedge{}  (\muparrow{}P[x]))

Date html generated: 2017_10_01-AM-08_54_12
Last ObjectModification: 2017_07_26-PM-04_35_57

Theory : bags

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