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406
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1 {-# OPTIONS --cubical-compatible --safe #-}
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2
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59
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3 module logic where
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4
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5 open import Level
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6 open import Relation.Nullary
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141
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7 open import Relation.Binary hiding (_⇔_)
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59
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8 open import Data.Empty
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9
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10
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11 data Bool : Set where
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12 true : Bool
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13 false : Bool
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14
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15 record _∧_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
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16 field
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17 proj1 : A
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18 proj2 : B
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19
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20 data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
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21 case1 : A → A ∨ B
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22 case2 : B → A ∨ B
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23
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330
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24 -- data ⊥ : Set where
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25
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26 -- ⊥-elim : {n : Level} {A : Set n } → ⊥ → A
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27 --⊥-elim ()
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28
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29 -- ¬_ : {n : Level } → Set n → Set n
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30 -- ¬ A = A → ⊥
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31
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59
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32 _⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m ) → Set (n ⊔ m)
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33 _⇔_ A B = ( A → B ) ∧ ( B → A )
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34
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35 contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A
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36 contra-position {n} {m} {A} {B} f ¬b a = ¬b ( f a )
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37
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38 double-neg : {n : Level } {A : Set n} → A → ¬ ¬ A
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39 double-neg A notnot = notnot A
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40
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41 double-neg2 : {n : Level } {A : Set n} → ¬ ¬ ¬ A → ¬ A
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42 double-neg2 notnot A = notnot ( double-neg A )
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43
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44 de-morgan : {n : Level } {A B : Set n} → A ∧ B → ¬ ( (¬ A ) ∨ (¬ B ) )
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45 de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and ))
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46 de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and ))
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47
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48 dont-or : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ A → B
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49 dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a )
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50 dont-or {A} {B} (case2 b) ¬A = b
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51
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52 dont-orb : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ B → A
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53 dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b )
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54 dont-orb {A} {B} (case1 a) ¬B = a
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55
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56
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57 infixr 130 _∧_
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58 infixr 140 _∨_
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59 infixr 150 _⇔_
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60
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