refinements to Basics

src/Basics0.lagda

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+
+---
+title     : "Basics: Functional Programming in Agda"
+layout    : page
+permalink : /Basics
+---
+
+\begin{code}
+open import Data.Empty       using (⊥; ⊥-elim)
+open import Relation.Nullary using (¬_; Dec; yes; no)
+open import Relation.Binary.PropositionalEquality
+                             using (_≡_; refl; _≢_; trans; sym)
+\end{code}
+
+# Natural numbers
+
+\begin{code}
+data ℕ : Set where
+  zero : ℕ
+  suc : ℕ → ℕ
+{-# BUILTIN NATURAL ℕ #-}
+\end{code}
+
+\begin{code}
+congruent : ∀ {m n} → m ≡ n → suc m ≡ suc n
+congruent refl = refl
+
+injective : ∀ {m n} → suc m ≡ suc n → m ≡ n
+injective refl = refl
+
+distinct : ∀ {m} → zero ≢ suc m
+distinct ()
+\end{code}
+
+\begin{code}
+_≟_ : ∀ (m n : ℕ) → Dec (m ≡ n)
+zero ≟ zero =  yes refl
+zero ≟ suc n =  no (λ()) 
+suc m ≟ zero =  no (λ())
+suc m ≟ suc n with m ≟ n
+... | yes refl =  yes refl
+... | no p =  no (λ r → p (injective r))
+\end{code}
+
+# Addition and its properties
+
+\begin{code}
+_+_ : ℕ → ℕ → ℕ
+zero + n = n
+suc m + n = suc (m + n)
+\end{code}
+
+\begin{code}
++-assoc : ∀ m n p → (m + n) + p ≡ m + (n + p)
++-assoc zero n p = refl
++-assoc (suc m) n p rewrite +-assoc m n p = refl
+
++-zero : ∀ m → m + zero ≡ m
++-zero zero = refl
++-zero (suc m) rewrite +-zero m = refl
+
++-suc : ∀ m n → m + (suc n) ≡ suc (m + n)
++-suc zero n = refl
++-suc (suc m) n rewrite +-suc m n = refl
+
++-comm : ∀ m n → m + n ≡ n + m
++-comm m zero =  +-zero m
++-comm m (suc n) rewrite +-suc m n | +-comm m n = refl
+\end{code}
+
+# Equality and decidable equality for naturals
+
+
+
+
+# Showing `double` injective
+
+\begin{code}
+double : ℕ → ℕ
+double zero  =  zero
+double (suc n)  =  suc (suc (double n))
+\end{code}
+
+\begin{code}
+double-injective : ∀ m n → double m ≡ double n → m ≡ n
+double-injective zero zero refl = refl
+double-injective zero (suc n) ()
+double-injective (suc m) zero ()
+double-injective (suc m) (suc n) r =
+   congruent (double-injective m n (injective (injective r)))
+\end{code}
+
+In Coq, the inductive proof for `double-injective`
+is sensitive to how one inducts on `m` and `n`. In Agda, that aspect
+is straightforward. However, Agda requires helper functions for
+injection and congruence which are not required in Coq.
+
+I can remove the use of `congruent` by rewriting with its argument.
+Is there an easy way to remove the two uses of `injective`?

src/extra/Basics-old.lagda

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+---
+title     : "Basics: Functional Programming in Agda"
+layout    : page
+permalink : /Basics
+---
+
+\begin{code}
+open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+\end{code}
+
+The functional programming style brings programming closer to
+simple, everyday mathematics: If a procedure or method has no side
+effects, then (ignoring efficiency) all we need to understand
+about it is how it maps inputs to outputs -- that is, we can think
+of it as just a concrete method for computing a mathematical
+function.  This is one sense of the word "functional" in
+"functional programming."  The direct connection between programs
+and simple mathematical objects supports both formal correctness
+proofs and sound informal reasoning about program behavior.
+
+The other sense in which functional programming is "functional" is
+that it emphasizes the use of functions (or methods) as
+_first-class_ values -- i.e., values that can be passed as
+arguments to other functions, returned as results, included in
+data structures, etc.  The recognition that functions can be
+treated as data in this way enables a host of useful and powerful
+idioms.
+
+Other common features of functional languages include _algebraic
+data types_ and _pattern matching_, which make it easy to
+construct and manipulate rich data structures, and sophisticated
+_polymorphic type systems_ supporting abstraction and code reuse.
+Agda shares all of these features.
+
+This chapter introduces the most essential elements of Agda.
+
+## Enumerated Types
+
+One unusual aspect of Agda is that its set of built-in
+features is _extremely_ small. For example, instead of providing
+the usual palette of atomic data types (booleans, integers,
+strings, etc.), Agda offers a powerful mechanism for defining new
+data types from scratch, from which all these familiar types arise
+as instances.
+
+Naturally, the Agda distribution comes with an extensive standard
+library providing definitions of booleans, numbers, and many
+common data structures like lists and hash tables.  But there is
+nothing magic or primitive about these library definitions.  To
+illustrate this, we will explicitly recapitulate all the
+definitions we need in this course, rather than just getting them
+implicitly from the library.
+
+To see how this definition mechanism works, let's start with a
+very simple example.
+
+### Days of the Week
+
+The following declaration tells Agda that we are defining
+a new set of data values -- a _type_.
+
+\begin{code}
+data Day : Set where
+  monday    : Day
+  tuesday   : Day
+  wednesday : Day
+  thursday  : Day
+  friday    : Day
+  saturday  : Day
+  sunday    : Day
+\end{code}
+
+The type is called `day`, and its members are `monday`,
+`tuesday`, etc.  The second and following lines of the definition
+can be read "`monday` is a `day`, `tuesday` is a `day`, etc."
+
+Having defined `day`, we can write functions that operate on
+days.
+
+\begin{code}
+nextWeekday : Day -> Day
+nextWeekday monday    = tuesday
+nextWeekday tuesday   = wednesday
+nextWeekday wednesday = thursday
+nextWeekday thursday  = friday
+nextWeekday friday    = monday
+nextWeekday saturday  = monday
+nextWeekday sunday    = monday
+\end{code}
+
+One thing to note is that the argument and return types of
+this function are explicitly declared.  Like most functional
+programming languages, Agda can often figure out these types for
+itself when they are not given explicitly -- i.e., it performs
+_type inference_ -- but we'll include them to make reading
+easier.
+
+Having defined a function, we should check that it works on
+some examples. There are actually three different ways to do this
+in Agda.
+
+First, we can use the Emacs command `C-c C-n` to evaluate a
+compound expression involving `nextWeekday`. For instance, `nextWeekday
+friday` should evaluate to `monday`. If you have a computer handy, this
+would be an excellent moment to fire up Agda and try this for yourself.
+Load this file, `Basics.lagda`, load it using `C-c C-l`, submit the
+above example to Agda, and observe the result.
+
+Second, we can record what we _expect_ the result to be in the
+form of an Agda type:
+
+\begin{code}
+test-nextWeekday : nextWeekday (nextWeekday saturday) ≡ tuesday
+\end{code}
+
+This declaration does two things: it makes an assertion (that the second
+weekday after `saturday` is `tuesday`), and it gives the assertion a name
+that can be used to refer to it later.
+
+Having made the assertion, we must also verify it. We do this by giving
+a term of the above type:
+
+\begin{code}
+test-nextWeekday = refl
+\end{code}
+
+There is no essential difference between the definition for
+`test-nextWeekday` here and the definition for `nextWeekday` above,
+except for the new symbol for equality `≡` and the constructor `refl`.
+The details of these are not important for now (we'll come back to them in
+a bit), but essentially `refl` can be read as "The assertion we've made
+can be proved by observing that both sides of the equality evaluate to the
+same thing, after some simplification."
+
+Third, we can ask Agda to _compile_ some program involving our definition,
+This facility is very interesting, since it gives us a way to construct
+_fully certified_ programs. We'll come back to this topic in later chapters.