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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