Nuprl Lemma : mapfilter-as-reduce
∀[L,p,f:Top]. (mapfilter(λx.f[x];λx.p[x];L) ~ reduce(λx,a. if p[x] then [f[x] / a] else a fi ;[];L))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
reduce: reduce(f;k;as),
cons: [a / b],
nil: [],
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
lambda: λx.A[x],
sqequal: s ~ t
Definitions unfolded in proof :
mapfilter: mapfilter(f;P;L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
filter: filter(P;l),
reduce: reduce(f;k;as),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
strict1: strict1(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
map: map(f;as),
list_ind: list_ind,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T,
guard: {T},
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
top: Top
Lemmas referenced :
top_wf,
map_nil_lemma,
sqle_wf_base,
map_cons_lemma,
map-ifthenelse,
is-exception_wf,
base_wf,
has-value_wf_base,
sqequal-list_ind
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueCallbyvalue,
hypothesis,
callbyvalueReduce,
callbyvalueExceptionCases,
inlFormation,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inrFormation,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
divergentSqle,
sqleRule,
sqleReflexivity,
because_Cache,
sqequalAxiom
Latex:
\mforall{}[L,p,f:Top]. (mapfilter(\mlambda{}x.f[x];\mlambda{}x.p[x];L) \msim{} reduce(\mlambda{}x,a. if p[x] then [f[x] / a] else a fi ;[];L)\000C)
Date html generated:
2016_05_15-PM-03_56_34
Last ObjectModification:
2016_01_16-AM-10_58_31
Theory : general
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