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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