### Nuprl Lemma : bag-summation-cons

`∀[R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R].`
`  ∀[T:Type]. ∀[f:T ⟶ R]. ∀[b:bag(T)]. ∀[a:T].  (Σ(x∈a.b). f[x] = (f[a] add Σ(x∈b). f[x]) ∈ R) `
`  supposing IsMonoid(R;add;zero) ∧ Comm(R;add)`

Proof

Definitions occuring in Statement :  bag-summation: `Σ(x∈b). f[x]` cons-bag: `x.b` bag: `bag(T)` comm: `Comm(T;op)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` infix_ap: `x f y` so_apply: `x[s]` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T` monoid_p: `IsMonoid(T;op;id)`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` single-bag: `{x}` bag-append: `as + bs` cons-bag: `x.b` append: `as @ bs` all: `∀x:A. B[x]` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` prop: `ℙ` cand: `A c∧ B` so_apply: `x[s]` so_lambda: `λ2x.t[x]` true: `True` squash: `↓T` infix_ap: `x f y` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` monoid_p: `IsMonoid(T;op;id)`
Lemmas referenced :  list_ind_cons_lemma list_ind_nil_lemma bag_wf monoid_p_wf comm_wf single-bag_wf bag-summation_wf infix_ap_wf equal_wf squash_wf true_wf bag-summation-append bag-summation-single iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin sqequalRule extract_by_obid dependent_functionElimination isect_memberEquality voidElimination voidEquality hypothesis hypothesisEquality isectElimination axiomEquality because_Cache cumulativity functionEquality universeEquality productEquality functionExtensionality applyEquality equalityTransitivity equalitySymmetry independent_pairFormation lambdaEquality independent_isectElimination natural_numberEquality imageElimination imageMemberEquality baseClosed independent_functionElimination

Latex:
\mforall{}[R:Type].  \mforall{}[add:R  {}\mrightarrow{}  R  {}\mrightarrow{}  R].  \mforall{}[zero:R].
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  R].  \mforall{}[b:bag(T)].  \mforall{}[a:T].    (\mSigma{}(x\mmember{}a.b).  f[x]  =  (f[a]  add  \mSigma{}(x\mmember{}b).  f[x]))