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213
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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
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6 open import Data.Empty
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7
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387
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8 data One {n : Level } : Set n where
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9 OneObj : One
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10
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11 data Two : Set where
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12 i0 : Two
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13 i1 : Two
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213
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14
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15 data Bool : Set where
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16 true : Bool
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17 false : Bool
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18
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19 record _∧_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
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422
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20 constructor ⟪_,_⟫
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213
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21 field
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22 proj1 : A
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23 proj2 : B
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24
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25 data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
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26 case1 : A → A ∨ B
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27 case2 : B → A ∨ B
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28
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29 _⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m ) → Set (n ⊔ m)
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30 _⇔_ A B = ( A → B ) ∧ ( B → A )
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31
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32 contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A
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33 contra-position {n} {m} {A} {B} f ¬b a = ¬b ( f a )
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34
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35 double-neg : {n : Level } {A : Set n} → A → ¬ ¬ A
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36 double-neg A notnot = notnot A
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37
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38 double-neg2 : {n : Level } {A : Set n} → ¬ ¬ ¬ A → ¬ A
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39 double-neg2 notnot A = notnot ( double-neg A )
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40
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41 de-morgan : {n : Level } {A B : Set n} → A ∧ B → ¬ ( (¬ A ) ∨ (¬ B ) )
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42 de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and ))
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43 de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and ))
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44
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45 dont-or : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ A → B
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46 dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a )
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47 dont-or {A} {B} (case2 b) ¬A = b
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48
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49 dont-orb : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ B → A
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50 dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b )
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51 dont-orb {A} {B} (case1 a) ¬B = a
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52
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53
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54 infixr 130 _∧_
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55 infixr 140 _∨_
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56 infixr 150 _⇔_
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57
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