### Nuprl Lemma : reduce-as-accum

`∀[T,A:Type]. ∀[f:T ⟶ A ⟶ A].`
`  ∀[L:T List]. ∀[a:A].`
`    (reduce(f;a;L) = accumulate (with value p and list item x): f x pover list:  Lwith starting value: a) ∈ A) `
`  supposing ∀x,y:T. ∀a:A.  ((f x (f y a)) = (f y (f x a)) ∈ A)`

Proof

Definitions occuring in Statement :  reduce: `reduce(f;k;as)` list_accum: list_accum list: `T List` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` so_apply: `x[s]` implies: `P `` Q` all: `∀x:A. B[x]` top: `Top` squash: `↓T` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q`
Lemmas referenced :  list_induction all_wf equal_wf reduce_wf list_accum_wf list_wf reduce_nil_lemma list_accum_nil_lemma reduce_cons_lemma list_accum_cons_lemma squash_wf true_wf iff_weakening_equal
Rules used in proof :  cut thin introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity functionExtensionality applyEquality hypothesis independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality lambdaFormation rename imageElimination equalityTransitivity equalitySymmetry because_Cache natural_numberEquality imageMemberEquality baseClosed universeEquality independent_isectElimination productElimination functionEquality isect_memberFormation axiomEquality

Latex:
\mforall{}[T,A:Type].  \mforall{}[f:T  {}\mrightarrow{}  A  {}\mrightarrow{}  A].
\mforall{}[L:T  List].  \mforall{}[a:A].
(reduce(f;a;L)
=  accumulate  (with  value  p  and  list  item  x):
f  x  p
over  list:
L
with  starting  value:
a))
supposing  \mforall{}x,y:T.  \mforall{}a:A.    ((f  x  (f  y  a))  =  (f  y  (f  x  a)))

Date html generated: 2017_04_17-AM-08_03_06
Last ObjectModification: 2017_02_27-PM-04_33_44

Theory : list_1

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