### Nuprl Lemma : bag-map-combine

`∀[A,B,C:Type]. ∀[g:A ⟶ bag(B)]. ∀[f:B ⟶ C]. ∀[bs:bag(A)].  (bag-map(f;⋃x∈bs.g[x]) = ⋃x∈bs.bag-map(f;g[x]) ∈ bag(C))`

Proof

Definitions occuring in Statement :  bag-combine: `⋃x∈bs.f[x]` bag-map: `bag-map(f;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` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` uimplies: `b supposing a` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` empty-bag: `{}` top: `Top` single-bag: `{x}` bag-append: `as + bs` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` true: `True` squash: `↓T` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  bag_wf list_wf quotient-member-eq permutation_wf permutation-equiv equal_wf bag-map_wf bag-combine_wf list-subtype-bag equal-wf-base list_induction bag_combine_empty_lemma bag_map_empty_lemma empty-bag_wf list_ind_cons_lemma list_ind_nil_lemma top_wf single-bag_wf subtype_rel_bag bag-append_wf squash_wf true_wf bag-combine-single-left bag-map-append bag-combine-append-left iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality sqequalRule isect_memberEquality axiomEquality because_Cache functionEquality universeEquality pointwiseFunctionalityForEquality pertypeElimination productElimination equalityTransitivity equalitySymmetry lambdaFormation rename lambdaEquality independent_isectElimination dependent_functionElimination independent_functionElimination hyp_replacement applyLambdaEquality functionExtensionality applyEquality productEquality voidElimination voidEquality equalityUniverse levelHypothesis natural_numberEquality imageElimination imageMemberEquality baseClosed

Latex:
\mforall{}[A,B,C:Type].  \mforall{}[g:A  {}\mrightarrow{}  bag(B)].  \mforall{}[f:B  {}\mrightarrow{}  C].  \mforall{}[bs:bag(A)].
(bag-map(f;\mcup{}x\mmember{}bs.g[x])  =  \mcup{}x\mmember{}bs.bag-map(f;g[x]))

Date html generated: 2017_10_01-AM-08_47_33
Last ObjectModification: 2017_07_26-PM-04_32_01

Theory : bags

Home Index