-- 8th Summer School on Formal Techniques -- Menlo College, Atherton, California, US -- -- 21-25 May 2018 -- -- Lecture 1: Introduction to Agda -- -- File 1: The Curry-Howard Isomorphism {-# OPTIONS --allow-unsolved-metas #-} module Prelude where -- Natural numbers as our first example of -- an inductive data type. data ℕ : Set where zero : ℕ suc : (n : ℕ) → ℕ -- To use decimal notation for numerals, like -- 2 instead of (suc (suc zero)), connect it -- to the built-in natural numbers. {-# BUILTIN NATURAL ℕ #-} -- Lists are a parameterized inductive data type. data List (A : Set) : Set where [] : List A _∷_ : (x : A) (xs : List A) → List A -- \ : : infixr 6 _∷_ -- C-c C-l load -- Disjoint sum type. data _⊎_ (S T : Set) : Set where -- \uplus inl : S → S ⊎ T inr : T → S ⊎ T infixr 4 _⊎_ -- The empty sum is the type with 0 alternatives, -- which is the empty type. -- By the Curry-Howard-Isomorphism, -- which views a proposition as the set/type of its proofs, -- it is also the absurd proposition. data False : Set where ⊥-elim : False → {A : Set} → A ⊥-elim () {A} -- C-c C-SPC give -- C-c C-, show hypotheses and goal -- C-c C-. show hypotheses and infers type -- Tuple types -- The generic record type with two fields -- where the type of the second depends on the value of the first -- is called Sigma-type (or dependent sum), in analogy to -- -- Σ ℕ A = Σ A(n) = A(0) + A(1) + ... -- n ∈ ℕ record Σ (A : Set) (B : A → Set) : Set where -- \GS \Sigma constructor _,_ field fst : A snd : B fst open Σ -- module Σ {A : Set} {B : A → Set} (p : Σ A B) where -- fst : A -- fst = case p of (a , b) -> a -- snd : B fst -- snd = case p of (a , b) -> b infixr 5 _,_ -- The non-dependent version is the ordinary Cartesian product. _×_ : (S T : Set) → Set S × T = Σ S λ _ → T infixr 5 _×_ -- The record type with no fields has exactly one inhabitant -- namely the empty tuple record{} -- By Curry-Howard, it corresponds to Truth, as -- no evidence is needed to construct this proposition. record True : Set where test : True test = _ -- C-c C-= show constraints -- C-c C-r refine -- C-c C-s solve -- C-c C-SPC give -- C-c C-a auto -- Example: distributivity A ∧ (B ∨ C) → (A ∧ B) ∨ (A ∧ C) dist : ∀{A B C : Set} → A × (B ⊎ C) → (A × B) ⊎ (A × C) dist (a , inl b) = inl (a , b) dist (a , inr c) = inr (a , c) dist' : ∀{A B : Set} → A × (B ⊎ A) → (A × B) ⊎ (A × A) dist' (a , inl b) = inl (a , b) dist' (a , inr c) = inr (c , c) -- Relations -- Type-theoretically, the type of relations 𝓟(A×A) is -- -- A × A → Prop -- -- which is -- -- A × A → Set -- -- by the Curry-Howard-Isomorphism -- and -- -- A → A → Set -- -- by currying. Rel : (A : Set) → Set₁ Rel A = A → A → Set -- Less-or-equal on natural numbers _≤_ : Rel ℕ zero ≤ y = True suc x ≤ zero = False suc x ≤ suc y = x ≤ y -- C-c C-l load -- C-c C-c case split -- C-c C-, show goal and assumptions -- C-c C-. show goal and assumptions and current term -- C-c C-SPC give