--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/BAlgbra.agda	Tue Dec 31 11:22:52 2019 +0900
@@ -0,0 +1,107 @@
+open import Level
+open import Ordinals
+module BAlgbra {n : Level } (O : Ordinals {n})   where
+
+open import zf
+open import logic
+import OD 
+
+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 _∧_
+open _∨_
+open Bool
+
+_∩_ : ( A B : OD  ) → OD
+A ∩ B = record { def = λ x → def A x ∧ def B x } 
+
+_∪_ : ( A B : OD  ) → OD
+A ∪ B = record { def = λ x → def A x ∨ def B x } 
+
+_＼_ : ( A B : OD  ) → OD
+A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) }
+
+∪-Union : { A B : OD } → Union (A , B) ≡ ( A ∪ B )
+∪-Union {A} {B} = ==→o≡ ( record { eq→ =  lemma1 ; eq← = lemma2 } )  where
+    lemma1 :  {x : Ordinal} → def (Union (A , B)) x → def (A ∪ B) x
+    lemma1 {x} lt = lemma3 lt where
+        lemma4 : {y : Ordinal} → def (A , B) y ∧ def (ord→od y) x → ¬ (¬ ( def A x ∨ def B x) )
+        lemma4 {y} z with proj1 z
+        lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → def k x ) oiso (proj2 z)) )
+        lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → def k x ) oiso (proj2 z)) )
+        lemma3 : (((u : Ordinals.ord O) → ¬ def (A , B) u ∧ def (ord→od u) x) → ⊥) → def (A ∪ B) x
+        lemma3 not = double-neg-eilm (FExists _ lemma4 not)   -- choice
+    lemma2 :  {x : Ordinal} → def (A ∪ B) x → def (Union (A , B)) x
+    lemma2 {x} (case1 A∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A
+       (record { proj1 = case1 refl ; proj2 = subst (λ k → def A k) (sym diso) A∋x}))
+    lemma2 {x} (case2 B∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B
+       (record { proj1 = case2 refl ; proj2 = subst (λ k → def B k) (sym diso) B∋x}))
+
+∩-Select : { A B : OD } →  Select A (  λ x → ( A ∋ x ) ∧ ( B ∋ x )  ) ≡ ( A ∩ B )
+∩-Select {A} {B} = ==→o≡ ( record { eq→ =  lemma1 ; eq← = lemma2 } ) where
+    lemma1 : {x : Ordinal} → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → def (A ∩ B) x
+    lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → def B k ) diso (proj2 (proj2 lt)) }
+    lemma2 : {x : Ordinal} → def (A ∩ B) x → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x
+    lemma2 {x} lt = record { proj1 = proj1 lt ; proj2 =
+        record { proj1 = subst (λ k → def A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → def B k ) (sym diso) (proj2 lt) } }
+
+dist-ord : {p q r : OD } → p ∩ ( q ∪ r ) ≡   ( p ∩ q ) ∪ ( p ∩ r )
+dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
+    lemma1 :  {x : Ordinal} → def (p ∩ (q ∪ r)) x → def ((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} → def ((p ∩ q) ∪ (p ∩ r)) x → def (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 : OD } → p ∪ ( q ∩ r ) ≡   ( p ∪ q ) ∩ ( p ∪ r )
+dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
+    lemma1 : {x : Ordinal} → def (p ∪ (q ∩ r)) x → def ((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} → def ((p ∪ q) ∩ (p ∪ r)) x → def (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 -_ _++_ _**_
+