### Nuprl Lemma : bag-member-combine

`∀[A,B:Type]. ∀[bs:bag(A)]. ∀[f:A ⟶ bag(B)]. ∀[b:B].  uiff(b ↓∈ ⋃x∈bs.f[x];↓∃x:A. (x ↓∈ bs ∧ b ↓∈ f[x]))`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-combine: `⋃x∈bs.f[x]` bag: `bag(T)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` so_apply: `x[s]` exists: `∃x:A. B[x]` squash: `↓T` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` guard: `{T}` uiff: `uiff(P;Q)` squash: `↓T` bag-member: `x ↓∈ bs` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` empty-bag: `{}` bag-combine: `⋃x∈bs.f[x]` bag-size: `#(bs)` bag-map: `bag-map(f;bs)` bag-union: `bag-union(bbs)` concat: `concat(ll)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_stable: `SqStable(P)` length: `||as||` list_ind: list_ind bag-append: `as + bs` append: `as @ bs` single-bag: `{x}` cons: `[a / b]` nil: `[]` it: `⋅` true: `True` sq_or: `a ↓∨ b` cand: `A c∧ B`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf istype-less_than int_seg_properties int_seg_wf subtract-1-ge-0 decidable__equal_int subtract_wf subtype_base_sq set_subtype_base int_subtype_base intformnot_wf intformeq_wf itermSubtract_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_subtract_lemma decidable__le decidable__lt istype-le subtype_rel_self bag-cases itermAdd_wf int_term_value_add_lemma istype-nat bag-size_wf bag_wf istype-universe length_of_nil_lemma map_nil_lemma reduce_nil_lemma squash_wf false_wf bag-member_wf iff_weakening_uiff empty-bag_wf bag-member-empty-iff le_wf uiff_wf bag-combine_wf sq_stable__all sq_stable__uiff sq_stable__bag-member sq_stable__squash equal-wf-base subtract_nat_wf add-is-int-iff bag-combine-single-left true_wf bag-combine-append-left single-bag_wf iff_weakening_equal sq_or_wf bag-member-append bag-append_wf bag-member-single
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut thin lambdaFormation_alt extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType productElimination independent_pairEquality imageElimination imageMemberEquality baseClosed isectIsTypeImplies inhabitedIsType functionIsTypeImplies unionElimination applyEquality instantiate because_Cache equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality_alt productIsType hypothesis_subsumption addEquality functionIsType universeEquality productEquality promote_hyp hyp_replacement functionEquality intEquality equalityIstype pointwiseFunctionality baseApply closedConclusion inlFormation_alt unionEquality inrFormation_alt

Latex:
\mforall{}[A,B:Type].  \mforall{}[bs:bag(A)].  \mforall{}[f:A  {}\mrightarrow{}  bag(B)].  \mforall{}[b:B].
uiff(b  \mdownarrow{}\mmember{}  \mcup{}x\mmember{}bs.f[x];\mdownarrow{}\mexists{}x:A.  (x  \mdownarrow{}\mmember{}  bs  \mwedge{}  b  \mdownarrow{}\mmember{}  f[x]))

Date html generated: 2019_10_15-AM-11_01_38
Last ObjectModification: 2019_06_25-PM-03_26_03

Theory : bags

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