### Nuprl Lemma : bag-null-filter

`∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[b:bag(T)].  uiff(↑bag-null([x∈b|p[x]]);∀x:T. (x ↓∈ b `` (¬↑p[x])))`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-null: `bag-null(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]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` so_apply: `x[s]` prop: `ℙ` so_lambda: `λ2x.t[x]` implies: `P `` Q` subtype_rel: `A ⊆r B` uimplies: `b supposing a` empty-bag: `{}` bag-null: `bag-null(bs)` null: `null(as)` bag-filter: `[x∈b|p[x]]` filter: `filter(P;l)` reduce: `reduce(f;k;as)` list_ind: list_ind nil: `[]` it: `⋅` btrue: `tt` assert: `↑b` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` and: `P ∧ Q` not: `¬A` false: `False` true: `True` cons-bag: `x.b` top: `Top` bool: `𝔹` unit: `Unit` cons: `[a / b]` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` rev_uimplies: `rev_uimplies(P;Q)` squash: `↓T` sq_stable: `SqStable(P)` sq_or: `a ↓∨ b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  assert_wf bag-null_wf bag-filter_wf all_wf bag-member_wf not_wf squash_wf list_induction uiff_wf list-subtype-bag list_wf bag-member-empty-iff empty-bag_wf true_wf bag_filter_cons_lemma bool_wf eqtt_to_assert eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot assert-bag-null assert_witness bag_wf bag_to_squash_list sq_stable__uiff sq_stable_from_decidable decidable__assert sq_stable__all sq_stable__not bag-member-cons assert_elim and_wf not_assert_elim btrue_neq_bfalse iff_weakening_uiff equal-wf-T-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity hypothesisEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin dependent_functionElimination setEquality cumulativity applyEquality functionExtensionality hypothesis sqequalRule lambdaEquality functionEquality because_Cache independent_isectElimination independent_functionElimination independent_pairFormation isect_memberFormation lambdaFormation productElimination voidElimination natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry rename isect_memberEquality voidEquality unionElimination equalityElimination dependent_pairFormation promote_hyp instantiate universeEquality independent_pairEquality imageElimination hyp_replacement applyLambdaEquality imageMemberEquality baseClosed inlFormation addLevel levelHypothesis dependent_set_memberEquality setElimination isectEquality inrFormation

Latex:
\mforall{}[T:Type].  \mforall{}[p:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[b:bag(T)].    uiff(\muparrow{}bag-null([x\mmember{}b|p[x]]);\mforall{}x:T.  (x  \mdownarrow{}\mmember{}  b  {}\mRightarrow{}  (\mneg{}\muparrow{}p[x])))

Date html generated: 2017_10_01-AM-08_55_07
Last ObjectModification: 2017_07_26-PM-04_37_06

Theory : bags

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