### Nuprl Lemma : bag-map-union

`∀[T,S:Type]. ∀[f:T ⟶ bag(S)]. ∀[bbs:bag(bag(T))].`
`  (bag-map(f;bag-union(bbs)) = bag-union(bag-map(λb.bag-map(f;b);bbs)) ∈ bag(bag(S)))`

Proof

Definitions occuring in Statement :  bag-union: `bag-union(bbs)` bag-map: `bag-map(f;bs)` bag: `bag(T)` uall: `∀[x:A]. B[x]` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` bag: `bag(T)` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` nat: `ℕ` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` bag-union: `bag-union(bbs)` bag-map: `bag-map(f;bs)` concat: `concat(ll)` true: `True`
Lemmas referenced :  bag_wf list_wf permutation_wf equal_wf equal-wf-base nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma 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 less_than_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int map_nil_lemma reduce_nil_lemma map_cons_lemma reduce_cons_lemma map_append_sq bag-map_wf squash_wf true_wf bag-union_wf quotient-member-eq permutation-equiv
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution pointwiseFunctionalityForEquality extract_by_obid isectElimination thin cumulativity hypothesisEquality hypothesis sqequalRule pertypeElimination productElimination equalityTransitivity equalitySymmetry lambdaFormation because_Cache rename dependent_functionElimination independent_functionElimination productEquality isect_memberEquality axiomEquality functionEquality universeEquality setElimination intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll sqequalAxiom applyEquality unionElimination promote_hyp hypothesis_subsumption applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination imageMemberEquality

Latex:
\mforall{}[T,S:Type].  \mforall{}[f:T  {}\mrightarrow{}  bag(S)].  \mforall{}[bbs:bag(bag(T))].
(bag-map(f;bag-union(bbs))  =  bag-union(bag-map(\mlambda{}b.bag-map(f;b);bbs)))

Date html generated: 2017_10_01-AM-08_46_42
Last ObjectModification: 2017_07_26-PM-04_31_25

Theory : bags

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