### Nuprl Lemma : bag-bind-append

`∀[A,B:Type]. ∀[ba,bb:bag(A)]. ∀[f:A ⟶ bag(B)].  (bag-bind(ba + bb;f) = (bag-bind(ba;f) + bag-bind(bb;f)) ∈ bag(B))`

Proof

Definitions occuring in Statement :  bag-bind: `bag-bind(bs;f)` bag-append: `as + bs` bag: `bag(T)` uall: `∀[x:A]. B[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` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` so_apply: `x[s]` bag-bind: `bag-bind(bs;f)` bag-append: `as + bs` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` bag-map: `bag-map(f;bs)` bag-union: `bag-union(bbs)` concat: `concat(ll)` true: `True` squash: `↓T` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]`
Lemmas referenced :  bag_wf equal-wf-base list_wf permutation_wf equal_wf list_induction bag-bind_wf bag-append_wf list-subtype-bag list_ind_nil_lemma map_nil_lemma reduce_nil_lemma list_ind_cons_lemma map_cons_lemma reduce_cons_lemma bag-append-assoc2 squash_wf true_wf iff_weakening_equal quotient-member-eq permutation-equiv and_wf
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 rename hyp_replacement equalitySymmetry applyLambdaEquality productEquality because_Cache functionEquality isect_memberEquality axiomEquality universeEquality equalityTransitivity lambdaFormation dependent_functionElimination independent_functionElimination lambdaEquality applyEquality independent_isectElimination functionExtensionality voidElimination voidEquality equalityUniverse levelHypothesis natural_numberEquality imageElimination imageMemberEquality baseClosed dependent_set_memberEquality independent_pairFormation setElimination

Latex:
\mforall{}[A,B:Type].  \mforall{}[ba,bb:bag(A)].  \mforall{}[f:A  {}\mrightarrow{}  bag(B)].
(bag-bind(ba  +  bb;f)  =  (bag-bind(ba;f)  +  bag-bind(bb;f)))

Date html generated: 2017_10_01-AM-09_05_55
Last ObjectModification: 2017_07_26-PM-04_46_05

Theory : bags

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