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Step * 2 of Lemma filter_is_nil3


1. ∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹].  filter(P;L) ~ [] supposing (∀x∈L.¬↑P[x])
⊢ ∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)}  ⟶ 𝔹].  filter(P;L) ~ [] supposing (∀x∈L.¬↑P[x])
BY
{ (Auto THEN InstHyp [⌜{x:T| (x ∈ L)} ⌝;⌜L⌝;⌜P⌝] 1⋅ THEN Auto) }

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Latex:

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Latex:

1.  \mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].    filter(P;L)  \msim{}  []  supposing  (\mforall{}x\mmember{}L.\mneg{}\muparrow{}P[x])
\mvdash{}  \mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}].    filter(P;L)  \msim{}  []  supposing  (\mforall{}x\mmember{}L.\mneg{}\muparrow{}P[x])


By

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Latex:
(Auto  THEN  InstHyp  [\mkleeneopen{}\{x:T|  (x  \mmember{}  L)\}  \mkleeneclose{};\mkleeneopen{}L\mkleeneclose{};\mkleeneopen{}P\mkleeneclose{}]  1\mcdot{}  THEN  Auto)

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