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Step * of Lemma filter_wf4

∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[l:T List].  (filter(λx.P[x];l) ∈ {x:T| (x ∈ l) ∧ (↑P[x])}  List)
BY
{ (InductionOnList THEN Reduce 0 THEN Auto THEN SubsumeC ⌜{x:T| (x ∈ v)}  List⌝⋅ THEN Auto) }

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Latex:

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Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[l:T  List].    (filter(\mlambda{}x.P[x];l)  \mmember{}  \{x:T|  (x  \mmember{}  l)  \mwedge{}  (\muparrow{}P[x])\}    List)


By

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Latex:
(InductionOnList  THEN  Reduce  0  THEN  Auto  THEN  SubsumeC  \mkleeneopen{}\{x:T|  (x  \mmember{}  v)\}    List\mkleeneclose{}\mcdot{}  THEN  Auto)

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