### Nuprl Lemma : list_accum_is_reduce

`∀[A:Type]. ∀[f:A ⟶ A ⟶ A].`
`  (∀[as:A List]. ∀[n:A].`
`     (accumulate (with value a and list item b):`
`       f[a;b]`
`      over list:`
`        as`
`      with starting value:`
`       n)`
`     = reduce(f;n;as)`
`     ∈ A)) supposing `
`     (Assoc(A;λx,y. f[x;y]) and `
`     Comm(A;λx,y. f[x;y]))`

Proof

Definitions occuring in Statement :  reduce: `reduce(f;k;as)` list_accum: list_accum list: `T List` comm: `Comm(T;op)` assoc: `Assoc(T;op)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` 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` uimplies: `b supposing a` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` true: `True` squash: `↓T` so_lambda: `λ2x y.t[x; y]` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` assoc: `Assoc(T;op)` infix_ap: `x f y` comm: `Comm(T;op)`
Lemmas referenced :  list_wf assoc_wf comm_wf reduce_wf equal_wf squash_wf true_wf list_accum_as_reduce iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis hypothesisEquality sqequalRule sqequalHypSubstitution isect_memberEquality isectElimination thin axiomEquality because_Cache extract_by_obid cumulativity lambdaEquality applyEquality functionExtensionality equalityTransitivity equalitySymmetry functionEquality universeEquality lambdaFormation natural_numberEquality imageElimination independent_isectElimination imageMemberEquality baseClosed productElimination independent_functionElimination

Latex:
\mforall{}[A:Type].  \mforall{}[f:A  {}\mrightarrow{}  A  {}\mrightarrow{}  A].
(\mforall{}[as:A  List].  \mforall{}[n:A].
(accumulate  (with  value  a  and  list  item  b):
f[a;b]
over  list:
as
with  starting  value:
n)
=  reduce(f;n;as)))  supposing
(Assoc(A;\mlambda{}x,y.  f[x;y])  and
Comm(A;\mlambda{}x,y.  f[x;y]))

Date html generated: 2017_04_17-AM-07_38_25
Last ObjectModification: 2017_02_27-PM-04_11_28

Theory : list_1

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