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Step * 1 2 1 of Lemma dm-neg-neg


1. T : Type
2. eq : EqDecider(T)
3. x : Point(free-DeMorgan-lattice(T;eq))
4. ∀g,h:Hom(free-dist-lattice(T + T; union-deq(T;T;eq;eq));free-dist-lattice(T + T; union-deq(T;T;eq;eq))).
     (((λz.free-dl-inc(z))
      = (g o (λx.free-dl-inc(x)))
      ∈ ((T + T) ⟶ Point(free-dist-lattice(T + T; union-deq(T;T;eq;eq)))))
     ⇒ ((λz.free-dl-inc(z))
        = (h o (λx.free-dl-inc(x)))
        ∈ ((T + T) ⟶ Point(free-dist-lattice(T + T; union-deq(T;T;eq;eq)))))
     ⇒ (g = h ∈ Hom(free-dist-lattice(T + T; union-deq(T;T;eq;eq));free-dist-lattice(T + T; union-deq(T;T;eq;eq)))))
⊢ (λx.¬(x)) o (λx.¬(x)) ∈ Hom(free-dist-lattice(T + T; union-deq(T;T;eq;eq));free-dist-lattice(T + T;
                                                                                               union-deq(T;T;eq;eq)))
BY
{ (InstLemma `compose-bounded-lattice-hom` [⌜free-DeMorgan-lattice(T;eq)⌝;
   ⌜opposite-lattice(free-DeMorgan-lattice(T;eq))⌝;⌜free-DeMorgan-lattice(T;eq)⌝] ⋅
   THENA Auto
   ) }

1
1. T : Type
2. eq : EqDecider(T)
3. x : Point(free-DeMorgan-lattice(T;eq))
4. ∀g,h:Hom(free-dist-lattice(T + T; union-deq(T;T;eq;eq));free-dist-lattice(T + T; union-deq(T;T;eq;eq))).
     (((λz.free-dl-inc(z))
      = (g o (λx.free-dl-inc(x)))
      ∈ ((T + T) ⟶ Point(free-dist-lattice(T + T; union-deq(T;T;eq;eq)))))
     ⇒ ((λz.free-dl-inc(z))
        = (h o (λx.free-dl-inc(x)))
        ∈ ((T + T) ⟶ Point(free-dist-lattice(T + T; union-deq(T;T;eq;eq)))))
     ⇒ (g = h ∈ Hom(free-dist-lattice(T + T; union-deq(T;T;eq;eq));free-dist-lattice(T + T; union-deq(T;T;eq;eq)))))
5. ∀[f:Hom(free-DeMorgan-lattice(T;eq);opposite-lattice(free-DeMorgan-lattice(T;eq)))].
   ∀[g:Hom(opposite-lattice(free-DeMorgan-lattice(T;eq));free-DeMorgan-lattice(T;eq))].
     (g o f ∈ Hom(free-DeMorgan-lattice(T;eq);free-DeMorgan-lattice(T;eq)))
⊢ (λx.¬(x)) o (λx.¬(x)) ∈ Hom(free-dist-lattice(T + T; union-deq(T;T;eq;eq));free-dist-lattice(T + T;
                                                                                               union-deq(T;T;eq;eq)))

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Latex:

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Latex:

1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  x  :  Point(free-DeMorgan-lattice(T;eq))
4.  \mforall{}g,h:Hom(free-dist-lattice(T  +  T;  union-deq(T;T;eq;eq));free-dist-lattice(T  +  T;
                                                                                                                                                          union-deq(T;T;eq;eq))).
          (((\mlambda{}z.free-dl-inc(z))  =  (g  o  (\mlambda{}x.free-dl-inc(x))))
          {}\mRightarrow{}  ((\mlambda{}z.free-dl-inc(z))  =  (h  o  (\mlambda{}x.free-dl-inc(x))))
          {}\mRightarrow{}  (g  =  h))
\mvdash{}  (\mlambda{}x.\mneg{}(x))  o  (\mlambda{}x.\mneg{}(x))  \mmember{}  Hom(free-dist-lattice(T  +  T;  union-deq(T;T;eq;eq)
                                                                                                );free-dist-lattice(T  +  T;  union-deq(T;T;eq;eq)))


By

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Latex:
(InstLemma  `compose-bounded-lattice-hom`  [\mkleeneopen{}free-DeMorgan-lattice(T;eq)\mkleeneclose{};
  \mkleeneopen{}opposite-lattice(free-DeMorgan-lattice(T;eq))\mkleeneclose{};\mkleeneopen{}free-DeMorgan-lattice(T;eq)\mkleeneclose{}]  \mcdot{}
  THENA  Auto
  )

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