Nuprl Lemma : filter-as-mapfilter
∀[L,P:Top]. (filter(P;L) ~ mapfilter(λx.x;P;L))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
filter: filter(P;l),
uall: ∀[x:A]. B[x],
top: Top,
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,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
map: map(f;as),
list_ind: list_ind,
has-value: (a)↓,
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,
baseClosed,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
hypothesis,
baseApply,
closedConclusion,
hypothesisEquality,
inrFormation,
imageMemberEquality,
callbyvalueCallbyvalue,
callbyvalueReduce,
callbyvalueExceptionCases,
inlFormation,
imageElimination,
exceptionSqequal,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
divergentSqle,
sqleRule,
sqleReflexivity,
because_Cache,
sqequalAxiom
Latex:
\mforall{}[L,P:Top]. (filter(P;L) \msim{} mapfilter(\mlambda{}x.x;P;L))
Date html generated:
2016_05_14-PM-01_29_21
Last ObjectModification:
2016_01_15-AM-08_28_17
Theory : list_1
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