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Step * 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)))))
⊢ ¬(¬(x)) = x ∈ Point(free-DeMorgan-lattice(T;eq))
BY
{ (InstHyp [⌜λx.x⌝;⌜λx.¬(¬(x))⌝] (-1)⋅
THENM (ApFunToHypEquands `Z' ⌜Z x⌝ ⌜Point(free-DeMorgan-lattice(T;eq))⌝ (-1)⋅ THEN Reduce (-1) THEN Auto)
) }

1
.....wf..... 
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 ∈ Hom(free-dist-lattice(T + T; union-deq(T;T;eq;eq));free-dist-lattice(T + T; union-deq(T;T;eq;eq)))

2
.....wf..... 
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)) ∈ Hom(free-dist-lattice(T + T; union-deq(T;T;eq;eq));free-dist-lattice(T + T; union-deq(T;T;eq;eq)))

3
.....antecedent..... 
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)))))
⊢ (λz.free-dl-inc(z))
= ((λx.x) o (λx.free-dl-inc(x)))
∈ ((T + T) ⟶ Point(free-dist-lattice(T + T; union-deq(T;T;eq;eq))))

4
.....antecedent..... 
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)))))
⊢ (λz.free-dl-inc(z))
= ((λx.¬(¬(x))) o (λx.free-dl-inc(x)))
∈ ((T + T) ⟶ Point(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{}  \mneg{}(\mneg{}(x))  =  x


By

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Latex:
(InstHyp  [\mkleeneopen{}\mlambda{}x.x\mkleeneclose{};\mkleeneopen{}\mlambda{}x.\mneg{}(\mneg{}(x))\mkleeneclose{}]  (-1)\mcdot{}
THENM  (ApFunToHypEquands  `Z'  \mkleeneopen{}Z  x\mkleeneclose{}  \mkleeneopen{}Point(free-DeMorgan-lattice(T;eq))\mkleeneclose{}  (-1)\mcdot{}
              THEN  Reduce  (-1)
              THEN  Auto)
)

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