Nuprl Lemma : bag-null-filter

[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[b:bag(T)].  uiff(↑bag-null([x∈b|p[x]]);∀x:T. (x ↓∈  (¬↑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:  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: so_lambda: λ2x.t[x] implies:  Q subtype_rel: A ⊆B uimplies: 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 then else 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: ⇐⇒ Q rev_implies:  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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