Nuprl Lemma : mapfilter-as-accum
∀[A:Type]. ∀[L:A List]. ∀[p:A ⟶ 𝔹]. ∀[f:Top].
(mapfilter(f;p;L) ~ accumulate (with value a and list item x):
if p[x] then a @ [f[x]] else a fi
over list:
L
with starting value:
[]))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
append: as @ bs,
list_accum: list_accum,
cons: [a / b],
nil: [],
list: T List,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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]
Lemmas referenced :
mapfilter-as-accum-aux,
nil_wf,
top_wf,
list_ind_nil_lemma,
bool_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalAxiom,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[L:A List]. \mforall{}[p:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:Top].
(mapfilter(f;p;L) \msim{} accumulate (with value a and list item x):
if p[x] then a @ [f[x]] else a fi
over list:
L
with starting value:
[]))
Date html generated:
2016_05_15-PM-03_56_28
Last ObjectModification:
2015_12_27-PM-03_09_05
Theory : general
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