### 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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