--- a/src/logic.agda	Sat Feb 18 11:51:22 2023 +0900
+++ b/src/logic.agda	Tue Feb 21 12:02:41 2023 +0900
@@ -5,17 +5,15 @@
 open import Relation.Binary hiding (_⇔_ )
 open import Data.Empty
 
-data One {n : Level } : Set n where
-  OneObj : One
+
+data Bool : Set where
+    true : Bool
+    false : Bool
 
 data Two : Set where
    i0 : Two
    i1 : Two
 
-data Bool : Set where
-   true : Bool
-   false : Bool
-
 record  _∧_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
    constructor ⟪_,_⟫
    field
@@ -29,6 +27,13 @@
 _⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m )  → Set (n ⊔ m)
 _⇔_ A B =  ( A → B ) ∧ ( B → A )
 
+∧-exch : {n m : Level} {A  : Set n} { B : Set m } → A ∧ B → B ∧ A
+∧-exch p = ⟪ _∧_.proj2 p , _∧_.proj1 p ⟫
+
+∨-exch : {n m : Level} {A  : Set n} { B : Set m } → A ∨ B → B ∨ A
+∨-exch (case1 x) = case2 x
+∨-exch (case2 x) = case1 x
+
 contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A
 contra-position {n} {m} {A} {B}  f ¬b a = ¬b ( f a )
 
@@ -42,6 +47,10 @@
 de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and ))
 de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and ))
 
+de-morgan∨ : {n  : Level } {A B : Set n} →  A ∨ B  → ¬ ( (¬ A ) ∧ (¬ B ) )
+de-morgan∨ {n} {A} {B} (case1 a) and = ⊥-elim (  _∧_.proj1 and a )
+de-morgan∨ {n} {A} {B} (case2 b) and = ⊥-elim (  _∧_.proj2 and b )
+
 dont-or :　{n m : Level} {A  : Set n} { B : Set m } →  A ∨ B → ¬ A → B
 dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a )
 dont-or {A} {B} (case2 b) ¬A = b
@@ -50,6 +59,150 @@
 dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b )
 dont-orb {A} {B} (case1 a) ¬B = a
 
+infixr  130 _∧_
+infixr  140 _∨_
+infixr  150 _⇔_
+
+_/\_ : Bool → Bool → Bool 
+true /\ true = true
+_ /\ _ = false
+
+_\/_ : Bool → Bool → Bool 
+false \/ false = false
+_ \/ _ = true
+
+not : Bool → Bool 
+not true = false
+not false = true 
+
+_<=>_ : Bool → Bool → Bool  
+true <=> true = true
+false <=> false = true
+_ <=> _ = false
+
+open import Relation.Binary.PropositionalEquality
+
+not-not-bool : { b : Bool } → not (not b) ≡ b
+not-not-bool {true} = refl
+not-not-bool {false} = refl
+
+record Bijection {n m : Level} (R : Set n) (S : Set m) : Set (n Level.⊔ m)  where
+   field
+       fun←  :  S → R
+       fun→  :  R → S
+       fiso← : (x : R)  → fun← ( fun→ x )  ≡ x
+       fiso→ : (x : S ) → fun→ ( fun← x )  ≡ x
+
+injection :  {n m : Level} (R : Set n) (S : Set m) (f : R → S ) → Set (n Level.⊔ m)
+injection R S f = (x y : R) → f x ≡ f y → x ≡ y
+
+
+¬t=f : (t : Bool ) → ¬ ( not t ≡ t) 
+¬t=f true ()
+¬t=f false ()
+
+infixr  130 _\/_
+infixr  140 _/\_
+
+≡-Bool-func : {A B : Bool } → ( A ≡ true → B ≡ true ) → ( B ≡ true → A ≡ true ) → A ≡ B
+≡-Bool-func {true} {true} a→b b→a = refl
+≡-Bool-func {false} {true} a→b b→a with b→a refl
+... | ()
+≡-Bool-func {true} {false} a→b b→a with a→b refl
+... | ()
+≡-Bool-func {false} {false} a→b b→a = refl
+
+bool-≡-? : (a b : Bool) → Dec ( a ≡ b )
+bool-≡-? true true = yes refl
+bool-≡-? true false = no (λ ())
+bool-≡-? false true = no (λ ())
+bool-≡-? false false = yes refl
+
+¬-bool-t : {a : Bool} →  ¬ ( a ≡ true ) → a ≡ false
+¬-bool-t {true} ne = ⊥-elim ( ne refl )
+¬-bool-t {false} ne = refl
+
+¬-bool-f : {a : Bool} →  ¬ ( a ≡ false ) → a ≡ true
+¬-bool-f {true} ne = refl
+¬-bool-f {false} ne = ⊥-elim ( ne refl )
+
+¬-bool : {a : Bool} →  a ≡ false  → a ≡ true → ⊥
+¬-bool refl ()
+
+lemma-∧-0 : {a b : Bool} → a /\ b ≡ true → a ≡ false → ⊥
+lemma-∧-0 {true} {true} refl ()
+lemma-∧-0 {true} {false} ()
+lemma-∧-0 {false} {true} ()
+lemma-∧-0 {false} {false} ()
+
+lemma-∧-1 : {a b : Bool} → a /\ b ≡ true → b ≡ false → ⊥
+lemma-∧-1 {true} {true} refl ()
+lemma-∧-1 {true} {false} ()
+lemma-∧-1 {false} {true} ()
+lemma-∧-1 {false} {false} ()
+
+bool-and-tt : {a b : Bool} → a ≡ true → b ≡ true → ( a /\ b ) ≡ true
+bool-and-tt refl refl = refl
+
+bool-∧→tt-0 : {a b : Bool} → ( a /\ b ) ≡ true → a ≡ true 
+bool-∧→tt-0 {true} {true} refl = refl
+bool-∧→tt-0 {false} {_} ()
+
+bool-∧→tt-1 : {a b : Bool} → ( a /\ b ) ≡ true → b ≡ true 
+bool-∧→tt-1 {true} {true} refl = refl
+bool-∧→tt-1 {true} {false} ()
+bool-∧→tt-1 {false} {false} ()
+
+bool-or-1 : {a b : Bool} → a ≡ false → ( a \/ b ) ≡ b 
+bool-or-1 {false} {true} refl = refl
+bool-or-1 {false} {false} refl = refl
+bool-or-2 : {a b : Bool} → b ≡ false → (a \/ b ) ≡ a 
+bool-or-2 {true} {false} refl = refl
+bool-or-2 {false} {false} refl = refl
+
+bool-or-3 : {a : Bool} → ( a \/ true ) ≡ true 
+bool-or-3 {true} = refl
+bool-or-3 {false} = refl
+
+bool-or-31 : {a b : Bool} → b ≡ true  → ( a \/ b ) ≡ true 
+bool-or-31 {true} {true} refl = refl
+bool-or-31 {false} {true} refl = refl
+
+bool-or-4 : {a : Bool} → ( true \/ a ) ≡ true 
+bool-or-4 {true} = refl
+bool-or-4 {false} = refl
+
+bool-or-41 : {a b : Bool} → a ≡ true  → ( a \/ b ) ≡ true 
+bool-or-41 {true} {b} refl = refl
+
+bool-and-1 : {a b : Bool} →  a ≡ false → (a /\ b ) ≡ false
+bool-and-1 {false} {b} refl = refl
+bool-and-2 : {a b : Bool} →  b ≡ false → (a /\ b ) ≡ false
+bool-and-2 {true} {false} refl = refl
+bool-and-2 {false} {false} refl = refl
+
+
+open import Data.Nat
+open import Data.Nat.Properties
+
+_≥b_ : ( x y : ℕ) → Bool
+x ≥b y with <-cmp x y
+... | tri< a ¬b ¬c = false
+... | tri≈ ¬a b ¬c = true
+... | tri> ¬a ¬b c = true
+
+_>b_ : ( x y : ℕ) → Bool
+x >b y with <-cmp x y
+... | tri< a ¬b ¬c = false
+... | tri≈ ¬a b ¬c = false
+... | tri> ¬a ¬b c = true
+
+_≤b_ : ( x y : ℕ) → Bool
+x ≤b y  = y ≥b x
+
+_<b_ : ( x y : ℕ) → Bool
+x <b y  = y >b x
+
 open import Relation.Binary.PropositionalEquality
 
 ¬i0≡i1 : ¬ ( i0 ≡ i1 )
@@ -63,8 +216,3 @@
 ¬i1→i0 {i0} ne = refl
 ¬i1→i0 {i1} ne = ⊥-elim ( ne refl )
 
-
-infixr  130 _∧_
-infixr  140 _∨_
-infixr  150 _⇔_
-