### Nuprl Lemma : bag-combine-append-right

`∀[A,B:Type]. ∀[F,G:A ⟶ bag(B)]. ∀[ba:bag(A)].  (⋃x∈ba.F[x] + G[x] = (⋃x∈ba.F[x] + ⋃x∈ba.G[x]) ∈ bag(B))`

Proof

Definitions occuring in Statement :  bag-combine: `⋃x∈bs.f[x]` bag-append: `as + bs` 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` bag: `bag(T)` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` subtype_rel: `A ⊆r B` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bag-combine: `⋃x∈bs.f[x]` bag-append: `as + bs` bag-map: `bag-map(f;bs)` bag-union: `bag-union(bbs)` top: `Top` concat: `concat(ll)` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` empty-bag: `{}` squash: `↓T` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cand: `A c∧ B` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]`
Lemmas referenced :  bag_wf list_wf equal_wf bag-append_wf bag-combine_wf list-subtype-bag permutation_wf equal-wf-base list_induction map_nil_lemma reduce_nil_lemma list_ind_nil_lemma empty-bag_wf map_cons_lemma reduce_cons_lemma bag-append-assoc2 squash_wf true_wf bag-append-ac bag-append-comm iff_weakening_equal 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 hyp_replacement applyLambdaEquality applyEquality independent_isectElimination lambdaEquality functionExtensionality dependent_functionElimination independent_functionElimination productEquality isect_memberEquality axiomEquality functionEquality universeEquality voidElimination voidEquality imageElimination equalityUniverse levelHypothesis natural_numberEquality imageMemberEquality baseClosed independent_pairFormation

Latex:
\mforall{}[A,B:Type].  \mforall{}[F,G:A  {}\mrightarrow{}  bag(B)].  \mforall{}[ba:bag(A)].    (\mcup{}x\mmember{}ba.F[x]  +  G[x]  =  (\mcup{}x\mmember{}ba.F[x]  +  \mcup{}x\mmember{}ba.G[x]))

Date html generated: 2017_10_01-AM-08_47_20
Last ObjectModification: 2017_07_26-PM-04_31_55

Theory : bags

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