Nuprl Lemma : bag-intersection

[A:Type]. ∀[as,as',bs,bs':bag(A)].
  (↓∃x:A. (x ↓∈ as ∧ x ↓∈ bs)) supposing (#(as') < #(as) and #(bs') < #(bs) and ((as as') (bs bs') ∈ bag(A)))


Definitions occuring in Statement :  bag-member: x ↓∈ bs bag-size: #(bs) bag-append: as bs bag: bag(T) less_than: a < b uimplies: supposing a uall: [x:A]. B[x] exists: x:A. B[x] squash: T and: P ∧ Q universe: Type equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a squash: T exists: x:A. B[x] prop: subtype_rel: A ⊆B nat: bag-size: #(bs) bag-append: as bs bag: bag(T) quotient: x,y:A//B[x; y] and: P ∧ Q cand: c∧ B permutation: permutation(T;L1;L2) all: x:A. B[x] so_lambda: λ2x.t[x] so_apply: x[s] int_seg: {i..j-} lelt: i ≤ j < k top: Top ge: i ≥  decidable: Dec(P) or: P ∨ Q le: A ≤ B satisfiable_int_formula: satisfiable_int_formula(fmla) false: False implies:  Q not: ¬A less_than: a < b guard: {T} sq_type: SQType(T) true: True iff: ⇐⇒ Q rev_implies:  Q uiff: uiff(P;Q) inject: Inj(A;B;f) bool: 𝔹 unit: Unit it: btrue: tt ifthenelse: if then else fi  bfalse: ff bnot: ¬bb assert: b equipollent: B biject: Bij(A;B;f) less_than': less_than'(a;b)
Lemmas referenced :  bag_to_squash_list less_than_wf bag-size_wf equal_wf bag_wf bag-append_wf list-subtype-bag nat_wf permutation-length decidable__exists_int_seg length_wf int_seg_wf length-append non_neg_length length_append subtype_rel_list top_wf decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf itermAdd_wf intformle_wf itermConstant_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_formula_prop_eq_lemma int_formula_prop_wf lelt_wf member_wf list_wf append_wf permutation_wf squash_wf exists_wf bag-member_wf select_wf int_seg_properties decidable__le list-member-bag-member bag-member-select subtype_base_sq int_subtype_base length_wf_nat permute_list_select nat_properties iff_weakening_equal select_member l_member_wf true_wf select_append_front not_over_exists all_wf le_wf inject_wf decidable__equal_int le_weakening2 injection_le ifthenelse_wf lt_int_wf subtract_wf equipollent_functionality_wrt_equipollent equipollent-int_seg equipollent_weakening_ext-eq ext-eq_weakening bool_wf eqtt_to_assert assert_of_lt_int itermSubtract_wf int_term_value_subtract_lemma eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot compose_wf injection-composition int_seg_subtype false_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality imageElimination productElimination promote_hyp hypothesis equalitySymmetry hyp_replacement applyLambdaEquality cumulativity applyEquality because_Cache sqequalRule equalityTransitivity rename independent_isectElimination lambdaEquality setElimination pertypeElimination instantiate dependent_functionElimination natural_numberEquality functionExtensionality dependent_set_memberEquality independent_pairFormation isect_memberEquality voidElimination voidEquality unionElimination dependent_pairFormation int_eqEquality intEquality computeAll independent_functionElimination lambdaFormation imageMemberEquality baseClosed productEquality universeEquality addEquality functionEquality equalityElimination

\mforall{}[A:Type].  \mforall{}[as,as',bs,bs':bag(A)].
    (\mdownarrow{}\mexists{}x:A.  (x  \mdownarrow{}\mmember{}  as  \mwedge{}  x  \mdownarrow{}\mmember{}  bs))  supposing 
          (\#(as')  <  \#(as)  and 
          \#(bs')  <  \#(bs)  and 
          ((as  +  as')  =  (bs  +  bs')))

Date html generated: 2017_10_01-AM-08_59_39
Last ObjectModification: 2017_07_26-PM-04_41_42

Theory : bags

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