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Step * 1 1 1 1 of Lemma free-dl-functor_wf


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. X : Type
4. Y : Type
5. z : Type
6. f : X ⟶ Y
7. g : Y ⟶ z
8. x : X
9. (fdl-hom(free-dl(Y);λx.free-dl-generator(f x)) free-dl-generator(x)) = free-dl-generator(f x) ∈ Point(free-dl(Y))
10. (fdl-hom(free-dl(z);λx.free-dl-generator(g x)) free-dl-generator(f x))
= free-dl-generator(g (f x))
∈ Point(free-dl(z))
⊢ (fdl-hom(free-dl(z);λx@0.free-dl-generator(g (f x@0))) free-dl-generator(x))
= (fdl-hom(free-dl(z);λx.free-dl-generator(g x)) (fdl-hom(free-dl(Y);λx.free-dl-generator(f x)) free-dl-generator(x)))
∈ Point(free-dl(z))
BY
{ ((InstLemma `fdl-hom-agrees` [⌜X⌝;⌜free-dl(z)⌝;⌜λx@0.free-dl-generator(g (f x@0))⌝;⌜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. X : Type
4. Y : Type
5. z : Type
6. f : X ⟶ Y
7. g : Y ⟶ z
8. x : X
9. (fdl-hom(free-dl(Y);λx.free-dl-generator(f x)) free-dl-generator(x)) = free-dl-generator(f x) ∈ Point(free-dl(Y))
10. (fdl-hom(free-dl(z);λx.free-dl-generator(g x)) free-dl-generator(f x))
= free-dl-generator(g (f x))
∈ Point(free-dl(z))
11. (fdl-hom(free-dl(z);λx@0.free-dl-generator(g (f x@0))) free-dl-generator(x))
= free-dl-generator(g (f x))
∈ Point(free-dl(z))
⊢ (fdl-hom(free-dl(z);λx@0.free-dl-generator(g (f x@0))) free-dl-generator(x))
= (fdl-hom(free-dl(z);λx.free-dl-generator(g x)) (fdl-hom(free-dl(Y);λx.free-dl-generator(f x)) free-dl-generator(x)))
∈ Point(free-dl(z))

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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.  X  :  Type
4.  Y  :  Type
5.  z  :  Type
6.  f  :  X  {}\mrightarrow{}  Y
7.  g  :  Y  {}\mrightarrow{}  z
8.  x  :  X
9.  (fdl-hom(free-dl(Y);\mlambda{}x.free-dl-generator(f  x))  free-dl-generator(x))  =  free-dl-generator(f  x)
10.  (fdl-hom(free-dl(z);\mlambda{}x.free-dl-generator(g  x))  free-dl-generator(f  x))
=  free-dl-generator(g  (f  x))
\mvdash{}  (fdl-hom(free-dl(z);\mlambda{}x@0.free-dl-generator(g  (f  x@0)))  free-dl-generator(x))
=  (fdl-hom(free-dl(z);\mlambda{}x.free-dl-generator(g  x)) 
      (fdl-hom(free-dl(Y);\mlambda{}x.free-dl-generator(f  x))  free-dl-generator(x)))


By

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Latex:
((InstLemma  `fdl-hom-agrees`  [\mkleeneopen{}X\mkleeneclose{};\mkleeneopen{}free-dl(z)\mkleeneclose{};\mkleeneopen{}\mlambda{}x@0.free-dl-generator(g  (f  x@0))\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  Reduce  -1
  )

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