Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/logic.agda @ 923:85f6238a38db
use supf of zchain for (nmx : ¬ Maximal A ) → ⊥
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 19 Oct 2022 08:41:42 +0900 |
parents | a5f8084b8368 |
children | 375615f9d60f |
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module logic where open import Level open import Relation.Nullary open import Relation.Binary hiding (_⇔_ ) open import Data.Empty data One {n : Level } : Set n where OneObj : One 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 proj1 : A proj2 : B data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where case1 : A → A ∨ B case2 : B → A ∨ B _⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m ) → Set (n ⊔ m) _⇔_ A B = ( A → B ) ∧ ( B → A ) 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 ) double-neg : {n : Level } {A : Set n} → A → ¬ ¬ A double-neg A notnot = notnot A double-neg2 : {n : Level } {A : Set n} → ¬ ¬ ¬ A → ¬ A double-neg2 notnot A = notnot ( double-neg A ) de-morgan : {n : Level } {A B : Set n} → A ∧ B → ¬ ( (¬ A ) ∨ (¬ B ) ) de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and )) de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and )) 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 dont-orb : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ B → A dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b ) dont-orb {A} {B} (case1 a) ¬B = a infixr 130 _∧_ infixr 140 _∨_ infixr 150 _⇔_