### 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)))`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-size: `#(bs)` bag-append: `as + bs` bag: `bag(T)` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` squash: `↓T` and: `P ∧ Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` squash: `↓T` exists: `∃x:A. B[x]` prop: `ℙ` subtype_rel: `A ⊆r B` nat: `ℕ` bag-size: `#(bs)` bag-append: `as + bs` bag: `bag(T)` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` cand: `A 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 ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` implies: `P `` Q` not: `¬A` less_than: `a < b` guard: `{T}` sq_type: `SQType(T)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` inject: `Inj(A;B;f)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` equipollent: `A ~ 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

Latex:
\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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