Step
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1
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1
of Lemma
free-DeMorgan-algebra-property
1. T : Type@i'
2. eq : EqDecider(T)@i
3. dm : DeMorganAlgebra@i'
4. eq2 : EqDecider(Point(dm))@i
5. f : T ⟶ Point(dm)@i
6. g : Hom(free-DeMorgan-lattice(T;eq);dm)
7. (g 0) = 0 ∈ Point(dm)
8. (g 1) = 1 ∈ Point(dm)
9. (λp.case p of inl(a) => f a | inr(a) => ¬(f a)) = (g o (λx.free-dl-inc(x))) ∈ ((T + T) ⟶ Point(dm))
10. ∀i:T. ((g <i>) = (f i) ∈ Point(dm))
11. ∀i:T. ((g <1-i>) = ¬(f i) ∈ Point(dm))
12. ∀h:Hom(free-DeMorgan-lattice(T;eq);dm)
(((λp.case p of inl(a) => f a | inr(a) => ¬(f a)) = (g o (λx.free-dl-inc(x))) ∈ ((T + T) ⟶ Point(dm)))
⇒ ((λp.case p of inl(a) => f a | inr(a) => ¬(f a)) = (h o (λx.free-dl-inc(x))) ∈ ((T + T) ⟶ Point(dm)))
⇒ (g = h ∈ Hom(free-DeMorgan-lattice(T;eq);dm)))
13. ∀[x,y:Point(free-DeMorgan-lattice(T;eq))]. (¬(x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(free-DeMorgan-lattice(T;eq)))
14. ∀[x,y:Point(free-DeMorgan-lattice(T;eq))]. (¬(x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(free-DeMorgan-lattice(T;eq)))
15. ¬(0) = 1 ∈ Point(free-DeMorgan-lattice(T;eq))
16. ¬(1) = 0 ∈ Point(free-DeMorgan-lattice(T;eq))
⊢ λa.¬(g ¬(a)) ∈ Hom(free-DeMorgan-lattice(T;eq);dm)
BY
{ (DVar `g' THEN MemTypeCD THEN Reduce 0 THEN Auto) }
1
1. T : Type@i'
2. eq : EqDecider(T)@i
3. dm : DeMorganAlgebra@i'
4. eq2 : EqDecider(Point(dm))@i
5. f : T ⟶ Point(dm)@i
6. g : Point(free-DeMorgan-lattice(T;eq)) ⟶ Point(dm)
7. ∀[a,b:Point(free-DeMorgan-lattice(T;eq))].
((g a ∧ g b = (g a ∧ b) ∈ Point(dm)) ∧ (g a ∨ g b = (g a ∨ b) ∈ Point(dm)))
8. (g 0) = 0 ∈ Point(dm)
9. (g 1) = 1 ∈ Point(dm)
10. (λp.case p of inl(a) => f a | inr(a) => ¬(f a)) = (g o (λx.free-dl-inc(x))) ∈ ((T + T) ⟶ Point(dm))
11. ∀i:T. ((g <i>) = (f i) ∈ Point(dm))
12. ∀i:T. ((g <1-i>) = ¬(f i) ∈ Point(dm))
13. ∀h:Hom(free-DeMorgan-lattice(T;eq);dm)
(((λp.case p of inl(a) => f a | inr(a) => ¬(f a)) = (g o (λx.free-dl-inc(x))) ∈ ((T + T) ⟶ Point(dm)))
⇒ ((λp.case p of inl(a) => f a | inr(a) => ¬(f a)) = (h o (λx.free-dl-inc(x))) ∈ ((T + T) ⟶ Point(dm)))
⇒ (g = h ∈ Hom(free-DeMorgan-lattice(T;eq);dm)))
14. ∀[x,y:Point(free-DeMorgan-lattice(T;eq))]. (¬(x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(free-DeMorgan-lattice(T;eq)))
15. ∀[x,y:Point(free-DeMorgan-lattice(T;eq))]. (¬(x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(free-DeMorgan-lattice(T;eq)))
16. ¬(0) = 1 ∈ Point(free-DeMorgan-lattice(T;eq))
17. ¬(1) = 0 ∈ Point(free-DeMorgan-lattice(T;eq))
18. a : Point(free-DeMorgan-lattice(T;eq))
19. b : Point(free-DeMorgan-lattice(T;eq))
⊢ ¬(g ¬(a)) ∧ ¬(g ¬(b)) = ¬(g ¬(a ∧ b)) ∈ Point(dm)
2
1. T : Type@i'
2. eq : EqDecider(T)@i
3. dm : DeMorganAlgebra@i'
4. eq2 : EqDecider(Point(dm))@i
5. f : T ⟶ Point(dm)@i
6. g : Point(free-DeMorgan-lattice(T;eq)) ⟶ Point(dm)
7. ∀[a,b:Point(free-DeMorgan-lattice(T;eq))].
((g a ∧ g b = (g a ∧ b) ∈ Point(dm)) ∧ (g a ∨ g b = (g a ∨ b) ∈ Point(dm)))
8. (g 0) = 0 ∈ Point(dm)
9. (g 1) = 1 ∈ Point(dm)
10. (λp.case p of inl(a) => f a | inr(a) => ¬(f a)) = (g o (λx.free-dl-inc(x))) ∈ ((T + T) ⟶ Point(dm))
11. ∀i:T. ((g <i>) = (f i) ∈ Point(dm))
12. ∀i:T. ((g <1-i>) = ¬(f i) ∈ Point(dm))
13. ∀h:Hom(free-DeMorgan-lattice(T;eq);dm)
(((λp.case p of inl(a) => f a | inr(a) => ¬(f a)) = (g o (λx.free-dl-inc(x))) ∈ ((T + T) ⟶ Point(dm)))
⇒ ((λp.case p of inl(a) => f a | inr(a) => ¬(f a)) = (h o (λx.free-dl-inc(x))) ∈ ((T + T) ⟶ Point(dm)))
⇒ (g = h ∈ Hom(free-DeMorgan-lattice(T;eq);dm)))
14. ∀[x,y:Point(free-DeMorgan-lattice(T;eq))]. (¬(x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(free-DeMorgan-lattice(T;eq)))
15. ∀[x,y:Point(free-DeMorgan-lattice(T;eq))]. (¬(x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(free-DeMorgan-lattice(T;eq)))
16. ¬(0) = 1 ∈ Point(free-DeMorgan-lattice(T;eq))
17. ¬(1) = 0 ∈ Point(free-DeMorgan-lattice(T;eq))
18. a : Point(free-DeMorgan-lattice(T;eq))
19. b : Point(free-DeMorgan-lattice(T;eq))
20. ¬(g ¬(a)) ∧ ¬(g ¬(b)) = ¬(g ¬(a ∧ b)) ∈ Point(dm)
⊢ ¬(g ¬(a)) ∨ ¬(g ¬(b)) = ¬(g ¬(a ∨ b)) ∈ Point(dm)
Latex:
Latex:
1. T : Type@i'
2. eq : EqDecider(T)@i
3. dm : DeMorganAlgebra@i'
4. eq2 : EqDecider(Point(dm))@i
5. f : T {}\mrightarrow{} Point(dm)@i
6. g : Hom(free-DeMorgan-lattice(T;eq);dm)
7. (g 0) = 0
8. (g 1) = 1
9. (\mlambda{}p.case p of inl(a) => f a | inr(a) => \mneg{}(f a)) = (g o (\mlambda{}x.free-dl-inc(x)))
10. \mforall{}i:T. ((g <i>) = (f i))
11. \mforall{}i:T. ((g ə-i>) = \mneg{}(f i))
12. \mforall{}h:Hom(free-DeMorgan-lattice(T;eq);dm)
(((\mlambda{}p.case p of inl(a) => f a | inr(a) => \mneg{}(f a)) = (g o (\mlambda{}x.free-dl-inc(x))))
{}\mRightarrow{} ((\mlambda{}p.case p of inl(a) => f a | inr(a) => \mneg{}(f a)) = (h o (\mlambda{}x.free-dl-inc(x))))
{}\mRightarrow{} (g = h))
13. \mforall{}[x,y:Point(free-DeMorgan-lattice(T;eq))]. (\mneg{}(x \mwedge{} y) = \mneg{}(x) \mvee{} \mneg{}(y))
14. \mforall{}[x,y:Point(free-DeMorgan-lattice(T;eq))]. (\mneg{}(x \mvee{} y) = \mneg{}(x) \mwedge{} \mneg{}(y))
15. \mneg{}(0) = 1
16. \mneg{}(1) = 0
\mvdash{} \mlambda{}a.\mneg{}(g \mneg{}(a)) \mmember{} Hom(free-DeMorgan-lattice(T;eq);dm)
By
Latex:
(DVar `g' THEN MemTypeCD THEN Reduce 0 THEN Auto)
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