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1 module logic where
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2
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3 open import Level
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4 open import Relation.Nullary
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5 open import Relation.Binary hiding (_⇔_)
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6 open import Relation.Binary.PropositionalEquality
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7
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8 open import Data.Empty
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9 open import Data.Nat hiding (_⊔_)
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10
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11
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12 data Bool : Set where
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13 true : Bool
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14 false : Bool
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15
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16 record _∧_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
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17 constructor ⟪_,_⟫
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18 field
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19 proj1 : A
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20 proj2 : B
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21
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22 data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
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23 case1 : A → A ∨ B
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24 case2 : B → A ∨ B
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25
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26 _⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m ) → Set (n ⊔ m)
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27 _⇔_ A B = ( A → B ) ∧ ( B → A )
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28
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29 contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A
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30 contra-position {n} {m} {A} {B} f ¬b a = ¬b ( f a )
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31
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32 double-neg : {n : Level } {A : Set n} → A → ¬ ¬ A
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33 double-neg A notnot = notnot A
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34
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35 double-neg2 : {n : Level } {A : Set n} → ¬ ¬ ¬ A → ¬ A
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36 double-neg2 notnot A = notnot ( double-neg A )
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37
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38 de-morgan : {n : Level } {A B : Set n} → A ∧ B → ¬ ( (¬ A ) ∨ (¬ B ) )
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39 de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and ))
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40 de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and ))
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41
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42 dont-or : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ A → B
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43 dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a )
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44 dont-or {A} {B} (case2 b) ¬A = b
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45
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46 dont-orb : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ B → A
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47 dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b )
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48 dont-orb {A} {B} (case1 a) ¬B = a
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49
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50
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51 infixr 130 _∧_
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52 infixr 140 _∨_
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53 infixr 150 _⇔_
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54
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55 _/\_ : Bool → Bool → Bool
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56 true /\ true = true
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57 _ /\ _ = false
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58
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59 _<B?_ : ℕ → ℕ → Bool
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60 ℕ.zero <B? x = true
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61 ℕ.suc x <B? ℕ.zero = false
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62 ℕ.suc x <B? ℕ.suc xx = x <B? xx
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63
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64 -- _<BT_ : ℕ → ℕ → Set
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65 -- ℕ.zero <BT ℕ.zero = ⊤
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66 -- ℕ.zero <BT ℕ.suc b = ⊤
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67 -- ℕ.suc a <BT ℕ.zero = ⊥
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68 -- ℕ.suc a <BT ℕ.suc b = a <BT b
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69
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70
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71 _≟B_ : Decidable {A = Bool} _≡_
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72 true ≟B true = yes refl
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73 false ≟B false = yes refl
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74 true ≟B false = no λ()
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75 false ≟B true = no λ()
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76
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77 _\/_ : Bool → Bool → Bool
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78 false \/ false = false
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79 _ \/ _ = true
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80
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81 not_ : Bool → Bool
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82 not true = false
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83 not false = true
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84
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85 _<=>_ : Bool → Bool → Bool
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86 true <=> true = true
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87 false <=> false = true
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88 _ <=> _ = false
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89
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90 infixr 130 _\/_
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91 infixr 140 _/\_
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