Nuprl Lemma : bag-combine-as-accum

`∀[A,B:Type]. ∀[f:A ⟶ bag(B)]. ∀[bs:bag(A)].  (⋃b∈bs.f[b] = bag-accum(c,b.f[b] + c;{};bs) ∈ bag(B))`

Proof

Definitions occuring in Statement :  bag-accum: `bag-accum(v,x.f[v; x];init;bs)` bag-combine: `⋃x∈bs.f[x]` bag-append: `as + bs` empty-bag: `{}` bag: `bag(T)` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` squash: `↓T` exists: `∃x:A. B[x]` 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)` top: `Top` and: `P ∧ Q` prop: `ℙ` or: `P ∨ Q` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` subtype_rel: `A ⊆r B` bag-combine: `⋃x∈bs.f[x]` bag-union: `bag-union(bbs)` concat: `concat(ll)` reduce: `reduce(f;k;as)` list_ind: list_ind bag-map: `bag-map(f;bs)` map: `map(f;as)` empty-bag: `{}` bag-accum: `bag-accum(v,x.f[v; x];init;bs)` list_accum: list_accum cons-bag: `x.b` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bag-append: `as + bs`
Lemmas referenced :  bag_to_squash_list 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 list-cases product_subtype_list colength-cons-not-zero colength_wf_list istype-le list_wf subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le le_wf istype-nat equal_wf bag_wf bag-combine_wf bag-accum_wf empty-bag_wf bag-append_wf bag-append-assoc-comm istype-universe bag-combine-cons-left cons-bag_wf list-subtype-bag squash_wf true_wf subtype_rel_self iff_weakening_equal cons-bag-as-append bag-append-comm single-bag_wf list_accum_append subtype_rel_list top_wf bag-accum-single
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality imageElimination productElimination promote_hyp hypothesis rename lambdaFormation_alt setElimination 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 axiomEquality functionIsTypeImplies inhabitedIsType unionElimination hypothesis_subsumption equalityIstype because_Cache dependent_set_memberEquality_alt instantiate equalityTransitivity equalitySymmetry applyLambdaEquality baseApply closedConclusion baseClosed applyEquality intEquality sqequalBase hyp_replacement isectIsTypeImplies functionIsType universeEquality imageMemberEquality functionEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  bag(B)].  \mforall{}[bs:bag(A)].    (\mcup{}b\mmember{}bs.f[b]  =  bag-accum(c,b.f[b]  +  c;\{\};bs))

Date html generated: 2019_10_15-AM-11_00_34
Last ObjectModification: 2019_08_01-PM-01_36_17

Theory : bags

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