halfway through Quantifiers
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index.md
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index.md
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@ -26,7 +26,7 @@ material in a different way; see the [Preface](Preface).
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- [Equality: Equality and equational reasoning](Equality)
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- [Isomorphism: Isomorphism and embedding](Isomorphism)
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- [Connectives: Conjunction, Disjunction, and Implication](Connectives)
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- [Negation: Negation, with Classical and Intuitionistic Logic](Negation)
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- [Negation: Negation, with Intuitionistic and Classical Logic](Negation)
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- [Quantiers: Universals and Existentials](Quantifiers)
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- [Lists: Lists and other data types](Lists)
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- [Decidable: Booleans and decision procedures](Decidable)
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@ -534,7 +534,6 @@ Right identity follows from commutativity of sum and left identity.
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≃-∎
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\end{code}
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## Implication is function
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Given two propositions `A` and `B`, the implication `A → B` holds if
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@ -543,9 +542,9 @@ the function type, which has appeared throughout this book.
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Evidence that `A → B` holds is of the form
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λ{ x → N }
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λ (x : A) → N x
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where `N` is a term of type `B` containing as a free variable `x` of type `A`.
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where `N x` is a term of type `B` containing as a free variable `x` of type `A`.
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Given a term `L` providing evidence that `A → B` holds, and a term `M`
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providing evidence that `A` holds, the term `L M` provides evidence that
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`B` holds. In other words, evidence that `A → B` holds is a function that
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@ -555,7 +554,7 @@ Put another way, if we know that `A → B` and `A` both hold,
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then we may conclude that `B` holds.
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\begin{code}
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→-elim : ∀ {A B : Set} → (A → B) → A → B
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→-elim f x = f x
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→-elim L M = L M
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\end{code}
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In medieval times, this rule was known by the name *modus ponens*.
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It corresponds to function application.
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@ -566,10 +565,10 @@ while applying a function is referred to as *eliminating* the function.
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Elimination followed by introduction is the identity.
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\begin{code}
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η-→ : ∀ {A B : Set} (f : A → B) → (λ{x → f x}) ≡ f
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η-→ : ∀ {A B : Set} (f : A → B) → (λ (x : A) → f x) ≡ f
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η-→ {A} {B} f = extensionality η-helper
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where
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η-helper : (x : A) → (λ (x′ : A) → f x′) x ≡ f x
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η-helper : (x : A) → (λ (x : A) → f x) x ≡ f x
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η-helper x = refl
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\end{code}
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The proof depends on extensionality.
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@ -672,8 +671,8 @@ That is, the assertion that if either `A` holds or `B` holds then `C` holds
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is the same as the assertion that if `A` holds then `C` holds and if
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`B` holds then `C` holds. The proof of the left inverse requires extensionality.
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\begin{code}
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→-distributes-⊎ : ∀ {A B C : Set} → (A ⊎ B → C) ≃ ((A → C) × (B → C))
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→-distributes-⊎ =
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→-distrib-⊎ : ∀ {A B C : Set} → (A ⊎ B → C) ≃ ((A → C) × (B → C))
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→-distrib-⊎ =
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record
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{ to = λ{ f → ( (λ{ x → f (inj₁ x) }) , (λ{ y → f (inj₂ y) }) ) }
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; from = λ{ (g , h) → λ{ (inj₁ x) → g x ; (inj₂ y) → h y } }
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@ -695,8 +694,8 @@ is the same as the assertion that if `A` holds then `B` holds and if
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`A` holds then `C` holds. The proof of left inverse requires both extensionality
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and the rule `η-×` for products.
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\begin{code}
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→-distributes-× : ∀ {A B C : Set} → (A → B × C) ≃ ((A → B) × (A → C))
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→-distributes-× =
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→-distrib-× : ∀ {A B C : Set} → (A → B × C) ≃ ((A → B) × (A → C))
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→-distrib-× =
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record
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{ to = λ{ f → ( (λ{ x → proj₁ (f x) }) , (λ{ y → proj₂ (f y)}) ) }
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; from = λ{ (g , h) → λ{ x → (g x , h x) } }
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@ -711,8 +710,8 @@ and the rule `η-×` for products.
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Products distributes over sum, up to isomorphism. The code to validate
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this fact is similar in structure to our previous results.
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\begin{code}
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×-distributes-⊎ : ∀ {A B C : Set} → ((A ⊎ B) × C) ≃ ((A × C) ⊎ (B × C))
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×-distributes-⊎ =
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×-distrib-⊎ : ∀ {A B C : Set} → ((A ⊎ B) × C) ≃ ((A × C) ⊎ (B × C))
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×-distrib-⊎ =
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record
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{ to = λ { ((inj₁ x) , z) → (inj₁ (x , z))
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; ((inj₂ y) , z) → (inj₂ (y , z))
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@ -731,8 +730,8 @@ this fact is similar in structure to our previous results.
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Sums do not distribute over products up to isomorphism, but it is an embedding.
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\begin{code}
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⊎-distributes-× : ∀ {A B C : Set} → ((A × B) ⊎ C) ≲ ((A ⊎ C) × (B ⊎ C))
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⊎-distributes-× =
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⊎-distrib-× : ∀ {A B C : Set} → ((A × B) ⊎ C) ≲ ((A ⊎ C) × (B ⊎ C))
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⊎-distrib-× =
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record
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{ to = λ { (inj₁ (x , y)) → (inj₁ x , inj₁ y)
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; (inj₂ z) → (inj₂ z , inj₂ z)
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@ -800,6 +799,7 @@ This chapter uses the following unicode.
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⊎ U+228E MULTISET UNION (\u+)
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⊤ U+22A4 DOWN TACK (\top)
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⊥ U+22A5 UP TACK (\bot)
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η U+03B7 GREEK SMALL LETTER ETA (\eta)
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₁ U+2081 SUBSCRIPT ONE (\_1)
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₂ U+2082 SUBSCRIPT TWO (\_2)
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⇔ U+21D4 LEFT RIGHT DOUBLE ARROW (\<=>)
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@ -59,8 +59,9 @@ but on the left-hand side of the equation the argument has been instantiated to
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which requires that `x` and `y` are the same. Hence, for the right-hand side of the equation
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we need a term of type `x ≡ x`, and `refl` will do.
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It is instructive to develop `sym` interactively.
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To start, we supply a variable for the argument on the left, and a hole for the body on the right:
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It is instructive to develop `sym` interactively. To start, we supply
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a variable for the argument on the left, and a hole for the body on
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the right:
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sym : ∀ {ℓ} {A : Set ℓ} {x y : A} → x ≡ y → y ≡ x
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sym e = {! !}
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@ -75,8 +76,9 @@ If we go into the hole and type `C-C C-,` then Agda reports:
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.A : Set .ℓ
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.ℓ : .Agda.Primitive.Level
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If in the hole we type `C-C C-C e` then Agda will instantiate `e` to all possible constructors,
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with one equation for each. There is only one possible constructor:
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If in the hole we type `C-C C-C e` then Agda will instantiate `e` to
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all possible constructors, with one equation for each. There is only
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one possible constructor:
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sym : ∀ {ℓ} {A : Set ℓ} {x y : A} → x ≡ y → y ≡ x
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sym refl = {! !}
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@ -1,19 +1,16 @@
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---
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title : "Negation: Negation, with Classical and Intuitionistic Logic"
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title : "Negation: Negation, with Intuitionistic and Classical Logic"
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layout : page
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permalink : /Negation
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---
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This chapter introduces negation, and discusses intuitionistic
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and classical logic.
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## Imports
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\begin{code}
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import Relation.Binary.PropositionalEquality as Eq
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open Eq using (_≡_; refl; sym; trans; cong)
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open Eq.≡-Reasoning
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open import Isomorphism using (_≃_; ≃-sym; ≃-trans; _≲_)
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open Isomorphism.≃-Reasoning
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open import Data.Nat using (ℕ; zero; suc; _+_; _*_)
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open import Data.Nat.Properties.Simple using (+-suc)
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open import Data.Empty using (⊥; ⊥-elim)
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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open import Data.Product using (_×_; _,_; proj₁; proj₂)
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@ -84,11 +81,11 @@ can conclude `¬ ¬ A` (evidenced by `¬¬-intro x`). Then from
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`¬ ¬ ¬ A` and `¬ ¬ A` we have a contradiction (evidenced by
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`¬¬¬x (¬¬-intro x)`. Hence we have shown `¬ A`.
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Another law of logic is the *contrapositive*,
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Another law of logic is *contraposition*,
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stating that if `A` implies `B`, then `¬ B` implies `¬ A`.
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\begin{code}
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contrapositive : ∀ {A B : Set} → (A → B) → (¬ B → ¬ A)
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contrapositive f ¬y x = ¬y (f x)
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contraposition : ∀ {A B : Set} → (A → B) → (¬ B → ¬ A)
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contraposition f ¬y x = ¬y (f x)
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\end{code}
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Let `f` be evidence of `A → B` and let `¬y` be evidence of `¬ B`. We
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will show that assuming `A` leads to a contradiction, and hence
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@ -268,7 +265,7 @@ Philip Wadler, *International Conference on Functional Programming*, 2003.)
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### Exercise
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Prove the following four formulas are equivalent to each other,
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Prove the following three formulas are equivalent to each other,
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and to the formulas `EM` and `⊎-Dual-+` given earlier.
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\begin{code}
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¬¬-Elim Peirce Implication : Set₁
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@ -277,61 +274,34 @@ Peirce = ∀ {A B : Set} → (((A → B) → A) → A)
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Implication = ∀ {A B : Set} → (A → B) → ¬ A ⊎ B
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\end{code}
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<!--
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It turns out that an assertion such as `Set : Set` is paradoxical.
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Therefore, there are an infinite number of different levels of sets,
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where `Set lzero : Set lone` and `Set lone : Set ltwo`, and so on,
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where `lone` is `lsuc lzero`, and `ltwo` is `lsuc lone`, analogous to
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the way we represent the natural nuambers. So far, we have only used
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the type `Set`, which is equivalent to `Set lzero`. Here, since each
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of `double-negation` and the others defines a type, we need to use
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`Set₁` as the "type of types".
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-->
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[NOTES]
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Two halves of de Morgan's laws hold intuitionistically. The other two
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halves are each equivalent to the law of double negation.
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### Exercise (`¬-stable`, `×-stable`)
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Say that a formula is *stable* if double negation elimination holds for it.
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\begin{code}
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dem1 : ∀ {A B : Set} → A × B → ¬ (¬ A ⊎ ¬ B)
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dem1 (a , b) (inj₁ ¬a) = ¬a a
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dem1 (a , b) (inj₂ ¬b) = ¬b b
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Stable : Set → Set
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Stable A = ¬ ¬ A → A
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\end{code}
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Show that any negated formula is stable, and that the conjunction
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of two stable formulas is stable.
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\begin{code}
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¬-Stable : Set₁
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¬-Stable = ∀ {A : Set} → Stable (¬ A)
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dem2 : ∀ {A B : Set} → A ⊎ B → ¬ (¬ A × ¬ B)
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dem2 (inj₁ a) (¬a , ¬b) = ¬a a
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dem2 (inj₂ b) (¬a , ¬b) = ¬b b
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×-Stable : Set₁
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×-Stable = ∀ {A B : Set} → Stable A → Stable B → Stable (A × B)
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\end{code}
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For the other variant of De Morgan's law, one way is an isomorphism.
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## Standard Prelude
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Definitions similar to those in this chapter can be found in the standard library.
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\begin{code}
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-- dem-≃ : ∀ {A B : Set} → (¬ (A ⊎ B)) ≃ (¬ A × ¬ B)
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-- dem-≃ = →-distributes-⊎
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import Relation.Nullary using (¬_)
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import Relation.Nullary.Negation using (contraposition)
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\end{code}
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The other holds in only one direction.
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\begin{code}
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dem-half : ∀ {A B : Set} → ¬ A ⊎ ¬ B → ¬ (A × B)
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dem-half (inj₁ ¬a) (a , b) = ¬a a
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dem-half (inj₂ ¬b) (a , b) = ¬b b
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\end{code}
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The other variant does not appear to be equivalent to classical logic.
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So that undermines my idea that basic propositions are either true
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intuitionistically or equivalent to classical logic.
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For several of the laws equivalent to classical logic, the reverse
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direction holds in intuitionistic long.
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\begin{code}
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implication-inv : ∀ {A B : Set} → (¬ A ⊎ B) → A → B
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implication-inv (inj₁ ¬a) a = ⊥-elim (¬a a)
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implication-inv (inj₂ b) a = b
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demorgan-inv : ∀ {A B : Set} → A ⊎ B → ¬ (¬ A × ¬ B)
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demorgan-inv (inj₁ a) (¬a , ¬b) = ¬a a
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demorgan-inv (inj₂ b) (¬a , ¬b) = ¬b b
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\end{code}
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## Unicode
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This chapter uses the following unicode.
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¬ U+00AC NOT SIGN (\neg)
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@ -4,6 +4,8 @@ layout : page
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permalink : /Quantifiers
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---
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This chapter introduces universal and existential quatification.
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## Imports
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\begin{code}
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@ -16,6 +18,8 @@ open import Data.Nat using (ℕ; zero; suc; _+_; _*_)
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open import Data.Nat.Properties.Simple using (+-suc)
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open import Relation.Nullary using (¬_)
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open import Function using (_∘_)
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open import Data.Product using (_×_; _,_; proj₁; proj₂)
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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\end{code}
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In what follows, we will occasionally require [extensionality][extensionality].
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@ -29,52 +33,73 @@ postulate
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## Universals
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Given a variable `x` of type `A` and a proposition `B[x]` which
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Given a variable `x` of type `A` and a proposition `B x` which
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contains `x` as a free variable, the universally quantified
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proposition `∀ (x : A) → B[x]` holds if for every term `M` of type
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`A` the proposition `B[M]` holds. Here `B[M]` stands for
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the proposition `B[x]` with each free occurrence of `x` replaced by
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`M`. The variable `x` appears free in `B[x]` but bound in
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`∀ (x : A) → B[x]`. We formalise universal quantification using the
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proposition `∀ (x : A) → B x` holds if for every term `M` of type
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`A` the proposition `B M` holds. Here `B M` stands for
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the proposition `B x` with each free occurrence of `x` replaced by
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`M`. The variable `x` appears free in `B x` but bound in
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`∀ (x : A) → B x`. We formalise universal quantification using the
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dependent function type, which has appeared throughout this book.
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Evidence that `∀ (x : A) → B[x]` holds is of the form
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Evidence that `∀ (x : A) → B x` holds is of the form
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λ (x : A) → N[x]
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λ (x : A) → N x
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where `N[x]` is a term of type `B[x]`; here `N[x]` is a term and `B[x]`
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is a proposition (or type) both containing as a free variable `x` of
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type `A`. Given a term `L` providing evidence that `∀ (x : A) → B[x]`
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holds and a term `M` of type `A`, the term `L M` provides evidence
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that `B[M]` holds. In other words, evidence that `∀ (x : A) → B[x]`
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holds is a function that converts a term `M` of type `A` into evidence
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that `B[M]` holds.
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where `N x` is a term of type `B x`, and `N x` and `B x` both contain
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a free variable `x` of type `A`. Given a term `L` providing evidence
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that `∀ (x : A) → B x` holds, and a term `M` of type `A`, the term `L
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M` provides evidence that `B M` holds. In other words, evidence that
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`∀ (x : A) → B x` holds is a function that converts a term `M` of type
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`A` into evidence that `B M` holds.
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Put another way, if we know that `∀ (x : A) → B[x]` holds and that `M`
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is a term of type `A` then we may conclude that `B[M]` holds. In
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medieval times, this rule was known by the name *dictum de omni*.
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Put another way, if we know that `∀ (x : A) → B x` holds and that `M`
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is a term of type `A` then we may conclude that `B M` holds.
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\begin{code}
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∀-elim : ∀ {A : Set} {B : A → Set} → (∀ (x : A) → B x) → (M : A) → B M
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∀-elim L M = L M
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\end{code}
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In medieval times, this rule was known by the name *dictum de omni*.
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If we introduce a universal and the immediately eliminate it, we can
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always simplify the resulting term. Thus
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(λ (x : A) → N[x]) M
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simplifies to `N[M]` of type `B[M]`, where `N[M]` stands for the term
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`N[x]` with each free occurrence of `x` replaced by `M` of type `A`.
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Ordinary function types arise as the special case of dependent
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function types where the range does not depend on a variable drawn
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from the domain. When an ordinary function is viewed as evidence of
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implication, both its domain and range are viewed as types of
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evidence, whereas when a dependent function is viewed as evidence of a
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universal, its domain is viewed as a data type and its range is viewed
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as a type of evidence. This is just a matter of interpretation, since
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in Agda data types and types of evidence are indistinguishable.
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Unlike with conjunction, disjunction, and implication, there is no simple
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analogy between universals and arithmetic.
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### Exercise (`∀-distrib-×`)
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Show that universals distribute over conjunction.
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\begin{code}
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∀-Distrib-× = ∀ {A : Set} {B C : A → Set} →
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(∀ (x : A) → B x × C x) ≃ (∀ (x : A) → B x) × (∀ (x : A) → C x)
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\end{code}
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Compare this with the result (`→-distrib-×`) in Chapter [Connectives](Connectives).
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||||
### Exercise (`⊎∀-implies-∀⊎`)
|
||||
|
||||
Show that a disjunction of universals implies a universal of disjunctions.
|
||||
\begin{code}
|
||||
⊎∀-Implies-∀⊎ = ∀ {A : Set} { B C : A → Set } →
|
||||
(∀ (x : A) → B x) ⊎ (∀ (x : A) → C x) → ∀ (x : A) → B x ⊎ C x
|
||||
\end{code}
|
||||
Does the converse also hold? If so, prove; if not, explain why.
|
||||
|
||||
## Existentials
|
||||
|
||||
Given a variable `x` of type `A` and a proposition `B[x]` which
|
||||
Given a variable `x` of type `A` and a proposition `B x` which
|
||||
contains `x` as a free variable, the existentially quantified
|
||||
proposition `∃ (λ (x : A) → B[x])` holds if for some term `M` of type
|
||||
`A` the proposition `B[M]` holds. Here `B[M]` stands for
|
||||
the proposition `B[x]` with each free occurrence of `x` replaced by
|
||||
`M`. The variable `x` appears free in `B[x]` but bound in
|
||||
`∃ (λ (x : A) → B[x])`.
|
||||
proposition `∃[ x ] → B x` holds if for some term `M` of type
|
||||
`A` the proposition `B M` holds. Here `B M` stands for
|
||||
the proposition `B x` with each free occurrence of `x` replaced by
|
||||
`M`. The variable `x` appears free in `B x` but bound in
|
||||
`∃[ x ] → B x`.
|
||||
|
||||
It is common to adopt a notation such as `∃[ x : A ]→ B[x]`,
|
||||
which introduces `x` as a bound variable of type `A` that appears
|
||||
|
|
@ -85,18 +110,23 @@ to introduce `x` as a bound variable.
|
|||
We formalise existential quantification by declaring a suitable
|
||||
inductive type.
|
||||
\begin{code}
|
||||
data ∃ {A : Set} (B : A → Set) : Set where
|
||||
_,_ : (x : A) → B x → ∃ B
|
||||
data ∃ (A : Set) (B : A → Set) : Set where
|
||||
_,_ : (x : A) → B x → ∃ A B
|
||||
\end{code}
|
||||
Evidence that `∃ (λ (x : A) → B[x])` holds is of the form
|
||||
`(M , N)` where `M` is a term of type `A`, and `N` is evidence
|
||||
that `B[M]` holds.
|
||||
|
||||
\begin{code}
|
||||
syntax ∃ A (λ x → B) = ∃[ x ∈ A ] B
|
||||
\end{code}
|
||||
|
||||
Given evidence that `∃ (λ (x : A) → B[x])` holds, and
|
||||
given evidence that `∀ (x : A) → B[x] → C` holds, where `C` does
|
||||
not contain `x` as a free variable, we may conclude that `C` holds.
|
||||
\begin{code}
|
||||
∃-elim : ∀ {A : Set} {B : A → Set} {C : Set} → ∃ B → (∀ (x : A) → B x → C) → C
|
||||
∃-elim : ∀ {A : Set} {B : A → Set} {C : Set} →
|
||||
(∃[ x ∈ A ] B x) → (∀ (x : A) → B x → C) → C
|
||||
∃-elim (M , N) P = P M N
|
||||
\end{code}
|
||||
In other words, if we know for every `x` of type `A` that `B[x]`
|
||||
|
|
@ -148,7 +178,7 @@ allows us to simplify `m + suc n` to `suc (m + n)`.
|
|||
|
||||
Here is the proof in the forward direction.
|
||||
\begin{code}
|
||||
ev-ex : ∀ {n : ℕ} → even n → ∃(λ (m : ℕ) → 2 * m ≡ n)
|
||||
ev-ex : ∀ {n : ℕ} → even n → ∃[ m ∈ ℕ ] (2 * m ≡ n)
|
||||
ev-ex ev0 = (zero , refl)
|
||||
ev-ex (ev+2 ev) with ev-ex ev
|
||||
... | (m , refl) = (suc m , lemma m)
|
||||
|
|
@ -171,7 +201,7 @@ return a pair consisting of `suc m` and a proof that `2 * suc m ≡ suc
|
|||
|
||||
Here is the proof in the reverse direction.
|
||||
\begin{code}
|
||||
ex-ev : ∀ {n : ℕ} → ∃(λ (m : ℕ) → 2 * m ≡ n) → even n
|
||||
ex-ev : ∀ {n : ℕ} → ∃[ m ∈ ℕ ] (2 * m ≡ n) → even n
|
||||
ex-ev (zero , refl) = ev0
|
||||
ex-ev (suc m , refl) rewrite lemma m = ev+2 (ex-ev (m , refl))
|
||||
\end{code}
|
||||
|
|
@ -196,7 +226,7 @@ disjuntion and universals are generalised conjunction, this
|
|||
result is analogous to the one which tells us that negation
|
||||
of a disjuntion is isomorphic to a conjunction of negations.
|
||||
\begin{code}
|
||||
¬∃∀ : ∀ {A : Set} {B : A → Set} → (¬ ∃ (λ (x : A) → B x)) ≃ ∀ (x : A) → ¬ B x
|
||||
¬∃∀ : ∀ {A : Set} {B : A → Set} → (¬ ∃[ x ∈ A ] B x) ≃ ∀ (x : A) → ¬ B x
|
||||
¬∃∀ =
|
||||
record
|
||||
{ to = λ { ¬∃bx x bx → ¬∃bx (x , bx) }
|
||||
|
|
@ -223,7 +253,12 @@ requires extensionality.
|
|||
|
||||
*Exercise* Show `∃ (λ (x : A) → ¬ B x) → ¬ (∀ (x : A) → B x)`.
|
||||
|
||||
## Standard Prelude
|
||||
|
||||
Definitions similar to those in this chapter can be found in the standard library.
|
||||
\begin{code}
|
||||
import Data.Product using (∃)
|
||||
\end{code}
|
||||
|
||||
## Unicode
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue