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comparison logic.agda @ 213:22d435172d1a

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separate logic and nat
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 02 Aug 2019 12:17:10 +0900
parents
children 8b0715e28b33
comparison
equal deleted inserted replaced
212:0a1804cc9d0a 213:22d435172d1a
1 module logic where
2
3 open import Level
4 open import Relation.Nullary
5 open import Relation.Binary
6 open import Data.Empty
7
8
9 data Bool : Set where
10 true : Bool
11 false : Bool
12
13 record _∧_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
14 field
15 proj1 : A
16 proj2 : B
17
18 data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
19 case1 : A → A ∨ B
20 case2 : B → A ∨ B
21
22 _⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m ) → Set (n ⊔ m)
23 _⇔_ A B = ( A → B ) ∧ ( B → A )
24
25 contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A
26 contra-position {n} {m} {A} {B} f ¬b a = ¬b ( f a )
27
28 double-neg : {n : Level } {A : Set n} → A → ¬ ¬ A
29 double-neg A notnot = notnot A
30
31 double-neg2 : {n : Level } {A : Set n} → ¬ ¬ ¬ A → ¬ A
32 double-neg2 notnot A = notnot ( double-neg A )
33
34 de-morgan : {n : Level } {A B : Set n} → A ∧ B → ¬ ( (¬ A ) ∨ (¬ B ) )
35 de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and ))
36 de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and ))
37
38 dont-or :　{n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ A → B
39 dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a )
40 dont-or {A} {B} (case2 b) ¬A = b
41
42 dont-orb :　{n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ B → A
43 dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b )
44 dont-orb {A} {B} (case1 a) ¬B = a
45
46
47 infixr 130 _∧_
48 infixr 140 _∨_
49 infixr 150 _⇔_
50