### view BAlgbra.agda @ 396:8c092c042093

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author Shinji KONO Mon, 27 Jul 2020 15:11:54 +0900 6c72bee25653 44a484f17f26
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open import Level
open import Ordinals
module BAlgbra {n : Level } (O : Ordinals {n})   where

open import zf
open import logic
import OD
import ODC

open import Relation.Nullary
open import Relation.Binary
open import Data.Empty
open import Relation.Binary
open import Relation.Binary.Core
open import  Relation.Binary.PropositionalEquality
open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ )

open inOrdinal O
open OD O
open OD.OD
open ODAxiom odAxiom
open HOD

open _∧_
open _∨_
open Bool

--_∩_ : ( A B : HOD  ) → HOD
--A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } ;
--    odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y)) }

_∪_ : ( A B : HOD  ) → HOD
A ∪ B = record { od = record { def = λ x → odef A x ∨ odef B x } ;
odmax = omax (odmax A) (odmax B) ; <odmax = lemma } where
lemma :  {y : Ordinal} → odef A y ∨ odef B y → y o< omax (odmax A) (odmax B)
lemma {y} (case1 a) = ordtrans (<odmax A a) (omax-x _ _)
lemma {y} (case2 b) = ordtrans (<odmax B b) (omax-y _ _)

_＼_ : ( A B : HOD  ) → HOD
A ＼ B = record { od = record { def = λ x → odef A x ∧ ( ¬ ( odef B x ) ) }; odmax = odmax A ; <odmax = λ y → <odmax A (proj1 y) }

∪-Union : { A B : HOD } → Union (A , B) ≡ ( A ∪ B )
∪-Union {A} {B} = ==→o≡ ( record { eq→ =  lemma1 ; eq← = lemma2 } )  where
lemma1 :  {x : Ordinal} → odef (Union (A , B)) x → odef (A ∪ B) x
lemma1 {x} lt = lemma3 lt where
lemma4 : {y : Ordinal} → odef (A , B) y ∧ odef (ord→od y) x → ¬ (¬ ( odef A x ∨ odef B x) )
lemma4 {y} z with proj1 z
lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → odef k x ) oiso (proj2 z)) )
lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → odef k x ) oiso (proj2 z)) )
lemma3 : (((u : Ordinals.ord O) → ¬ odef (A , B) u ∧ odef (ord→od u) x) → ⊥) → odef (A ∪ B) x
lemma3 not = ODC.double-neg-eilm O (FExists _ lemma4 not)   -- choice
lemma2 :  {x : Ordinal} → odef (A ∪ B) x → odef (Union (A , B)) x
lemma2 {x} (case1 A∋x) = subst (λ k → odef (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A
(record { proj1 = case1 refl ; proj2 = d→∋ A A∋x } ))
lemma2 {x} (case2 B∋x) = subst (λ k → odef (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B
(record { proj1 = case2 refl ; proj2 = d→∋ B B∋x } ))

∩-Select : { A B : HOD } →  Select A (  λ x → ( A ∋ x ) ∧ ( B ∋ x )  ) ≡ ( A ∩ B )
∩-Select {A} {B} = ==→o≡ ( record { eq→ =  lemma1 ; eq← = lemma2 } ) where
lemma1 : {x : Ordinal} → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → odef (A ∩ B) x
lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → odef B k ) diso (proj2 (proj2 lt)) }
lemma2 : {x : Ordinal} → odef (A ∩ B) x → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x
lemma2 {x} lt = record { proj1 = proj1 lt ; proj2 =
record { proj1 = d→∋ A (proj1 lt) ; proj2 = d→∋ B (proj2 lt) } }

dist-ord : {p q r : HOD } → p ∩ ( q ∪ r ) ≡   ( p ∩ q ) ∪ ( p ∩ r )
dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
lemma1 :  {x : Ordinal} → odef (p ∩ (q ∪ r)) x → odef ((p ∩ q) ∪ (p ∩ r)) x
lemma1 {x} lt with proj2 lt
lemma1 {x} lt | case1 q∋x = case1 ( record { proj1 = proj1 lt ; proj2 = q∋x } )
lemma1 {x} lt | case2 r∋x = case2 ( record { proj1 = proj1 lt ; proj2 = r∋x } )
lemma2  : {x : Ordinal} → odef ((p ∩ q) ∪ (p ∩ r)) x → odef (p ∩ (q ∪ r)) x
lemma2 {x} (case1 p∩q) = record { proj1 = proj1 p∩q ; proj2 = case1 (proj2 p∩q ) }
lemma2 {x} (case2 p∩r) = record { proj1 = proj1 p∩r ; proj2 = case2 (proj2 p∩r ) }

dist-ord2 : {p q r : HOD } → p ∪ ( q ∩ r ) ≡   ( p ∪ q ) ∩ ( p ∪ r )
dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
lemma1 : {x : Ordinal} → odef (p ∪ (q ∩ r)) x → odef ((p ∪ q) ∩ (p ∪ r)) x
lemma1 {x} (case1 cp) = record { proj1 = case1 cp ; proj2 = case1 cp }
lemma1 {x} (case2 cqr) = record { proj1 = case2 (proj1 cqr) ; proj2 = case2 (proj2 cqr) }
lemma2 : {x : Ordinal} → odef ((p ∪ q) ∩ (p ∪ r)) x → odef (p ∪ (q ∩ r)) x
lemma2 {x} lt with proj1 lt | proj2 lt
lemma2 {x} lt | case1 cp | _ = case1 cp
lemma2 {x} lt | _ | case1 cp = case1 cp
lemma2 {x} lt | case2 cq | case2 cr = case2 ( record { proj1 = cq ; proj2 = cr } )

record IsBooleanAlgebra ( L : Set n)
( b1 : L )
( b0 : L )
( -_ : L → L  )
( _+_ : L → L → L )
( _*_ : L → L → L ) : Set (suc n) where
field
+-assoc : {a b c : L } → a + ( b + c ) ≡ (a + b) + c
*-assoc : {a b c : L } → a * ( b * c ) ≡ (a * b) * c
+-sym : {a b : L } → a + b ≡ b + a
-sym : {a b : L } → a * b  ≡ b * a
-aab : {a b : L } → a + ( a * b ) ≡ a
*-aab : {a b : L } → a * ( a + b ) ≡ a
-dist : {a b c : L } → a + ( b * c ) ≡ ( a * b ) + ( a * c )
*-dist : {a b c : L } → a * ( b + c ) ≡ ( a + b ) * ( a + c )
a+0 : {a : L } → a + b0 ≡ a
a*1 : {a : L } → a * b1 ≡ a
a+-a1 : {a : L } → a + ( - a ) ≡ b1
a*-a0 : {a : L } → a * ( - a ) ≡ b0

record BooleanAlgebra ( L : Set n) : Set (suc n) where
field
b1 : L
b0 : L
-_ : L → L
_++_ : L → L → L
_**_ : L → L → L
isBooleanAlgebra : IsBooleanAlgebra L b1 b0 -_ _++_ _**_

```