Nuprl Lemma : list_accum_permute
∀[T,A:Type]. ∀[g:T ⟶ A]. ∀[f:A ⟶ A ⟶ A].
(∀[as,bs:T List]. ∀[n:A].
(accumulate (with value a and list item z):
f[a;g[z]]
over list:
as @ bs
with starting value:
n)
= accumulate (with value a and list item z):
f[a;g[z]]
over list:
bs @ as
with starting value:
n)
∈ A)) supposing
(Assoc(A;λx,y. f[x;y]) and
Comm(A;λx,y. f[x;y]))
Proof
Definitions occuring in Statement :
append: as @ bs,
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],
so_apply: x[s],
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,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
implies: P ⇒ Q,
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],
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
uall_wf,
list_wf,
equal_wf,
list_accum_wf,
append_wf,
list_ind_nil_lemma,
append_nil_sq,
subtype_rel_list,
top_wf,
list_ind_cons_lemma,
list_accum_cons_lemma,
squash_wf,
true_wf,
cons_wf,
iff_weakening_equal,
assoc_wf,
comm_wf,
list_accum_append,
list_accum_permute1,
list_accum_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
because_Cache,
applyEquality,
functionExtensionality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
axiomEquality,
lambdaFormation,
rename,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
functionEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[g:T {}\mrightarrow{} A]. \mforall{}[f:A {}\mrightarrow{} A {}\mrightarrow{} A].
(\mforall{}[as,bs:T List]. \mforall{}[n:A].
(accumulate (with value a and list item z):
f[a;g[z]]
over list:
as @ bs
with starting value:
n)
= accumulate (with value a and list item z):
f[a;g[z]]
over list:
bs @ as
with starting value:
n))) 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_02
Last ObjectModification:
2017_02_27-PM-04_11_39
Theory : list_1
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