negation exercises

extra/answers/NegationAns.lagda

@@ -9,10 +9,11 @@ open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym)
 open import Data.Unit using (⊤; tt)
 open import Data.Empty using (⊥; ⊥-elim)
 open import Data.Sum using (_⊎_; inj₁; inj₂)
-open import Data.Product using (_×_; _,_; proj₁; proj₂)
+open import Data.Product using (_×_; proj₁; proj₂) renaming (_,_ to ⟨_,_⟩)
-open import Isomorphism using (_≃_)
+open import Relation.Nullary using (¬_)
-open import Negation using
+-- open import Isomorphism using (_≃_)
-  (¬_; id; EM; em-irrefutable; ¬¬-Elim; Peirce; Implication; ×-Implies-⊎)
+-- open import Negation using
+--   (¬_; id; EM; em-irrefutable; ¬¬-Elim; Peirce; Implication; ×-Implies-⊎)
 \end{code}
 
 In what follows, we occasionally require [extensionality][extensionality].
@@ -28,25 +29,66 @@ postulate
 *Equivalences for classical logic*
 
 \begin{code}
-ex1 : ¬¬-Elim → EM
+em-irrefutable : ∀ {A : Set} → ¬ ¬ (A ⊎ ¬ A)
-ex1 h = h em-irrefutable
+em-irrefutable k = k (inj₂ λ{ x → k (inj₁ x) })
 
-ex2 : EM → Implication
+EM : Set₁
-ex2 em f with em
+EM = ∀ {A : Set} → A ⊎ ¬ A
+
+¬¬-Elim : Set₁
+¬¬-Elim = ∀ {A : Set} → ¬ ¬ A → A
+
+Implication : Set₁
+Implication = ∀ {A B : Set} → (A → B) → ¬ A ⊎ B
+
+Peirce : Set₁
+Peirce = ∀ {A B : Set} → ((A → B) → A) → A
+
+DeMorgan : Set₁
+DeMorgan = ∀ {A B : Set} → ¬ (¬ A × ¬ B) → A ⊎ B
+
+ex1 : ¬¬-Elim → EM
+ex1 ¬¬-elim = ¬¬-elim em-irrefutable
+
+ex2 : EM → ¬¬-Elim
+ex2 em ¬¬a with em
+... | inj₁ a = a
+... | inj₂ ¬a = ⊥-elim (¬¬a ¬a)
+
+ex3 : EM → Implication
+ex3 em f with em
 ... | inj₁ a = inj₂ (f a)
 ... | inj₂ ¬a = inj₁ ¬a
 
-ex3 : EM → Peirce
+ex4 : Implication → EM
-ex3 em k with em
+ex4 imp = swap (imp (λ a → a))
+  where
+  swap : ∀ {A B} → A ⊎ B → B ⊎ A
+  swap (inj₁ a) = inj₂ a
+  swap (inj₂ b) = inj₁ b
+
+ex5 : EM → Peirce
+ex5 em k with em
 ... | inj₁ a = a
 ... | inj₂ ¬a = k (λ a → ⊥-elim (¬a a))
 
-ex4 : EM → ×-Implies-⊎
+ex6 : Peirce → EM
-ex4 em ¬a×b with em | em
+ex6 peirce = peirce (λ ¬em → ⊥-elim (em-irrefutable ¬em))
-... | inj₁ a | inj₁ b = ⊥-elim (¬a×b (a , b))
-... | inj₁ a | inj₂ ¬b = inj₂ ¬b
-... | inj₂ ¬a | _ = inj₁ ¬a
 
+ex7 : EM → DeMorgan
+ex7 em ¬[¬a׬b] with em | em
+... | inj₁ a | _ = inj₁ a
+... | inj₂ ¬a | inj₁ b = inj₂ b
+... | inj₂ ¬a | inj₂ ¬b = ⊥-elim (¬[¬a׬b] ⟨ ¬a , ¬b ⟩)
+
+ex8 : DeMorgan → EM
+ex8 deMorgan {A} = deMorgan ¬[¬a׬¬a]
+  where
+  ¬[¬a׬¬a] : ¬ (¬ A × ¬ ¬ A)
+  ¬[¬a׬¬a] ⟨ ¬a , ¬¬a ⟩ = ¬¬a ¬a
+
+
+{-
 ⊤⊥-iso : (¬ ⊥) ≃ ⊤
 ⊤⊥-iso =
   record
@@ -55,6 +97,7 @@ ex4 em ¬a×b with em | em
     ; from∘to =  λ{ _ → extensionality (λ ()) }
     ; to∘from =  λ{ tt → refl }
     }
+-}
 \end{code}  
 
 *Existentials and Universals*

src/plfa/Connectives.lagda

@@ -799,7 +799,7 @@ This is called a _weak distributive law_. Give the corresponding
 distributive law, and explain how it relates to the weak version.
 
 
-#### Exercise (`⊎×-implies-×⊎`)
+#### Exercise `⊎×-implies-×⊎`
 
 Show that a disjunct of conjuncts implies a conjunct of disjuncts.
 \begin{code}
@@ -809,7 +809,7 @@ postulate
 Does the converse hold? If so, prove; if not, give a counterexample.
 
 
-#### Exercise (`_⇔_`) {#iff}
+#### Exercise `_⇔_` {#iff}
 
 Define equivalence of propositions (also known as "if and only if") as follows.
 \begin{code}

src/plfa/Negation.lagda

@@ -156,14 +156,14 @@ the other.
 
 
 
-### Exercise (`≢`, `<-irrerflexive`)
+### Exercise `<-irrerflexive`
 
 Using negation, show that
 [strict inequality]({{ site.baseurl }}{% link out/plfa/Relations.md %}/#strict-inequality)
 is irreflexive, that is, `n < n` holds for no `n`.
 
 
-### Exercise (`trichotomy`)
+### Exercise `trichotomy`
 
 Show that strict inequality satisfies
 [trichotomy]({{ site.baseurl }}{% link out/plfa/Relations.md %}/#trichotomy),
@@ -177,21 +177,19 @@ Here "exactly one" means that one of the three must hold, and each implies the
 negation of the other two.
 
 
-### Exercise (`⊎-dual-×`)
+### Exercise `⊎-dual-×`
 
 Show that conjunction, disjunction, and negation are related by a
 version of De Morgan's Law.
-\begin{code}
+
-postulate
+    ¬ (A ⊎ B) ≃ (¬ A) × (¬ B)
-  ⊎-dual-× : ∀ {A B : Set} → ¬ (A ⊎ B) ≃ (¬ A) × (¬ B)
+
-\end{code}
 This result is an easy consequence of something we've proved previously.
 
 Is there also a term of the following type?
-\begin{code}
+
-postulate
+    ¬ (A × B) ≃ (¬ A) ⊎ (¬ B)
-  ×-dual-⊎ : ∀ {A B : Set} → ¬ (A × B) ≃ (¬ A) ⊎ (¬ B)
+
-\end{code}
 If so, prove; if not, explain why.
 
 
@@ -327,21 +325,20 @@ just handed to him.
 Philip Wadler, _International Conference on Functional Programming_, 2003.)
 
 
-#### Exercise
+#### Exercise `classical` (stretch)
 
-Consider the following five terms of the given types.
+Consider the following principles.
-\begin{code}
+
-postulate
+  * Excluded Middle: `A ⊎ ¬ A`, for all `A`
-  em′         : ∀ {A : Set} → A ⊎ ¬ A
+  * Double Negation Elimination: `¬ ¬ A → A`, for all `A`
-  ¬¬-elim     : ∀ {A : Set} → ¬ ¬ A → A
+  * Peirce's Law: `((A → B) → A) → A`, for all `A` and `B`.
-  peirce      : ∀ {A B : Set} → (((A → B) → A) → A)
+  * Implication as disjunction: `(A → B) → ¬ A ⊎ B`, for all `A` and `B`.
-  →-implies-⊎ : ∀ {A B : Set} → (A → B) → ¬ A ⊎ B
+  * De Morgan: `¬ (¬ A × ¬ B) → A ⊎ B`, for all `A` and `B`.
-  ×-implies-⊎ : ∀ {A B : Set} → ¬ (A × B) → (¬ A) ⊎ (¬ B)
+
-\end{code}
+Show that each of these implies all the others.
-Show that given any one term of the specified type, we can derive the others.
 
 
-### Exercise (`¬-stable`, `×-stable`)
+#### Exercise `stable` (stretch)
 
 Say that a formula is _stable_ if double negation elimination holds for it.
 \begin{code}
@@ -350,19 +347,6 @@ Stable A = ¬ ¬ A → A
 \end{code}
 Show that any negated formula is stable, and that the conjunction
 of two stable formulas is stable.
-\begin{code}
-postulate
-
-  ¬-stable : ∀ {A : Set}
-      ------------
-    → Stable (¬ A)
-
-  ×-stable : ∀ {A B : Set}
-    → Stable A
-    → Stable B
-      --------------
-    → Stable (A × B)
-\end{code}
 
 ## Standard Prelude