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Step * 3 1 1 1 of Lemma free-dist-lattice-adjunction


1. ∀Y:Type. (Point(free-dl(Y)) ~ free-dl-type(Y))
2. ∀Y:Type. ∀y:Y.  (free-dl-generator(y) ∈ Point(free-dl(Y)))
3. d : Type
4. x : d
⊢ (fdl-hom(free-dl(d);λg.g) 
   (fdl-hom(free-dl(Point(free-dl(d)));λx.free-dl-generator(free-dl-generator(x))) free-dl-generator(x)))
= free-dl-generator(x)
∈ Point(free-dl(d))
BY
{ ((InstLemma `fdl-hom-agrees` [⌜d⌝;⌜free-dl(Point(free-dl(d)))⌝;⌜λx.free-dl-generator(free-dl-generator(x))⌝;⌜x⌝]⋅
    THENA Auto
    )
   THEN Reduce -1
   ) }

1
1. ∀Y:Type. (Point(free-dl(Y)) ~ free-dl-type(Y))
2. ∀Y:Type. ∀y:Y.  (free-dl-generator(y) ∈ Point(free-dl(Y)))
3. d : Type
4. x : d
5. (fdl-hom(free-dl(Point(free-dl(d)));λx.free-dl-generator(free-dl-generator(x))) free-dl-generator(x))
= free-dl-generator(free-dl-generator(x))
∈ Point(free-dl(Point(free-dl(d))))
⊢ (fdl-hom(free-dl(d);λg.g) 
   (fdl-hom(free-dl(Point(free-dl(d)));λx.free-dl-generator(free-dl-generator(x))) free-dl-generator(x)))
= free-dl-generator(x)
∈ Point(free-dl(d))

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Latex:

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Latex:

1.  \mforall{}Y:Type.  (Point(free-dl(Y))  \msim{}  free-dl-type(Y))
2.  \mforall{}Y:Type.  \mforall{}y:Y.    (free-dl-generator(y)  \mmember{}  Point(free-dl(Y)))
3.  d  :  Type
4.  x  :  d
\mvdash{}  (fdl-hom(free-dl(d);\mlambda{}g.g) 
      (fdl-hom(free-dl(Point(free-dl(d)));\mlambda{}x.free-dl-generator(free-dl-generator(x))) 
        free-dl-generator(x)))
=  free-dl-generator(x)


By

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Latex:
((InstLemma  `fdl-hom-agrees`  [\mkleeneopen{}d\mkleeneclose{};\mkleeneopen{}free-dl(Point(free-dl(d)))\mkleeneclose{};
    \mkleeneopen{}\mlambda{}x.free-dl-generator(free-dl-generator(x))\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{}]\mcdot{}
    THENA  Auto
    )
  THEN  Reduce  -1
  )

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