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  1. using System;
  2. using System.Collections;
  3. using System.Collections.Generic;
  4.  
  5. /*
  6. A basic lens library for the purpose of demonstration.
  7. Implements a lens as the costate comonad coalgebra.
  8.  
  9. This library is not complete.
  10. A more complete lens library would take from
  11. Edward Kmett's to support polymorphic updates and traversal:
  12. http://hackage.haskell.org/package/lens
  13. */
  14. namespace Lens {
  15.  
  16. public struct And<A, B> {
  17.   public readonly A a;
  18.   public readonly B b;
  19.  
  20.   private And(A a, B b) {
  21.     this.a = a;
  22.     this.b = b;
  23.   }
  24.  
  25.   public And<X, B> First<X>(Func<A, X> f) {
  26.     return And<X, B>.and(f(a), b);
  27.   }
  28.  
  29.   public And<A, X> Second<X>(Func<B, X> f) {
  30.     return And<A, X>.and(a, f(b));
  31.   }
  32.  
  33.   public And<A, X> Select<X>(Func<B, X> f) {
  34.     return Second(f);
  35.   }
  36.  
  37.   public And<X, Y> BinarySelect<X, Y>(Func<A, X> f, Func<B, Y> g) {
  38.     return And<X, Y>.and(f(a), g(b));
  39.   }
  40.  
  41.   public C Apply<C>(Func<A, B, C> f) {
  42.     return f(a, b);
  43.   }
  44.  
  45.   public C ApplyFirst<C>(Func<A, C> f) {
  46.     return f(a);
  47.   }
  48.  
  49.   public C ApplySecond<C>(Func<B, C> f) {
  50.     return f(b);
  51.   }
  52.  
  53.   public X Fold<X>(Func<A, B, X> f) {
  54.     return f(a, b);
  55.   }
  56.  
  57.   public Store<B, A> ConstStore {
  58.     get {
  59.       var t = this;
  60.       return Store<B, A>.store(_ => t.a, t.b);
  61.     }
  62.   }
  63.  
  64.   public static And<A, B> and(A a, B b) {
  65.     return new And<A, B>(a, b);
  66.   }
  67. }
  68.  
  69. public struct Or<A, B> {
  70.   private readonly bool l;
  71.   private readonly A a;
  72.   private readonly B b;
  73.  
  74.   private Or(bool l, A a, B b) {
  75.     this.l = l;
  76.     this.a = a;
  77.     this.b = b;
  78.   }
  79.  
  80.   public bool IsLeft {
  81.     get {
  82.       return l;
  83.     }
  84.   }
  85.  
  86.   public bool IsRight {
  87.     get {
  88.       return !l;
  89.     }
  90.   }
  91.  
  92.   public Or<B, A> Swap {
  93.     get {
  94.       return l ? Or<B, A>.Right(a) : Or<B, A>.Left(b);
  95.     }
  96.   }
  97.  
  98.   public X Fold<X>(Func<A, X> left, Func<B, X> right) {
  99.     return l ? left(a) : right(b);
  100.   }
  101.  
  102.   public Or<A, X> Select<X>(Func<B, X> f) {
  103.     return Fold(a => Or<A, X>.Left(a), b => Or<A, X>.Right(f(b)));
  104.   }
  105.  
  106.   public Or<A, X> SelectMany<X>(Func<B, Or<A, X>> f) {
  107.     return Fold(a => Or<A, X>.Left(a), f);
  108.   }
  109.  
  110.   public Or<A, Y> SelectMany<X, Y>(Func<B, Or<A, X>> f, Func<B, X, Y> g) {
  111.     return SelectMany<Y>(b => f(b).Select<Y>(x => g(b, x)));
  112.   }
  113.  
  114.   public Or<X, Y> BinarySelect<X, Y>(Func<A, X> f, Func<B, Y> g) {
  115.     return Fold(a => Or<X, Y>.Left(f(a)), b => Or<X, Y>.Right(g(b)));
  116.   }
  117.  
  118.   public B RightValue(Func<A, B> f) {
  119.     return Fold(f, b => b);
  120.   }
  121.  
  122.   public A LeftValue(Func<B, A> f) {
  123.     return Fold(a => a, f);
  124.   }
  125.  
  126.   public B RightOr(Func<B> d) {
  127.     return RightValue(_ => d());
  128.   }
  129.  
  130.   public A LeftOr(Func<A> d) {
  131.     return LeftValue(_ => d());
  132.   }
  133.  
  134.   public void ForEachRight(Action<B> q) {
  135.     if(IsLeft)
  136.       q(b);
  137.   }
  138.  
  139.   public void ForEachLeft(Action<A> q) {
  140.     if(IsRight)
  141.       q(a);
  142.   }
  143.  
  144.   public static Or<A, B> Left(A a) {
  145.     return new Or<A, B>(true, a, default(B));
  146.   }
  147.  
  148.   public static Or<A, B> Right(B b) {
  149.     return new Or<A, B>(false, default(A), b);
  150.   }
  151. }
  152.  
  153. public class Store<A, B> {
  154.   public readonly Func<A, B> Put;
  155.   public readonly A Pos;
  156.  
  157.   private Store(Func<A, B> put, A pos) {
  158.     this.Put = put;
  159.     this.Pos = pos;
  160.   }
  161.  
  162.   public Store<A, C> Select<C>(Func<B, C> f) {
  163.     return Store<A, C>.store(a => f(Put(a)), Pos);
  164.   }
  165.  
  166.   /*
  167.   Store is a comonad.
  168.   */
  169.   public Store<A, C> CoSelectMany<C>(Func<Store<A, B>, C> f) {
  170.     return Store<A, C>.store(a => f(Store<A, B>.store(Put, a)), Pos);
  171.   }
  172.  
  173.   public B CoPoint {
  174.     get {
  175.       return Put(Pos);
  176.     }
  177.   }
  178.  
  179.   public Store<A, Store<A, B>> Duplicate {
  180.     get {
  181.       return Store<A, Store<A, B>>.store(a => Store<A, B>.store(Put, a), Pos);
  182.     }
  183.   }
  184.  
  185.   public Store<And<A, C>, And<B, D>> Product<C, D>(Store<C, D> s) {
  186.     return Store<And<A, C>, And<B, D>>.store(
  187.       x => And<B, D>.and(Put(x.a), s.Put(x.b))
  188.     , And<A, C>.and(Pos, s.Pos)
  189.     );
  190.   }
  191.  
  192.   public static Store<A, B> store(Func<A, B> put, A pos) {
  193.     return new Store<A, B>(put, pos);
  194.   }
  195. }
  196.  
  197. public struct Lens<A, B> {
  198.   private readonly Func<A, Store<B, A>> q;
  199.  
  200.   private Lens(Func<A, Store<B, A>> q) {
  201.     this.q = q;
  202.   }
  203.  
  204.   public Store<B, A> Run(A a) {
  205.     return q(a);
  206.   }
  207.  
  208.   public B Get(A a) {
  209.     return Run(a).Pos;
  210.   }
  211.  
  212.   public A Set(A a, B b) {
  213.     return Run(a).Put(b);
  214.   }
  215.  
  216.   public Func<A, A> Modify(Func<B, B> f) {
  217.     var t = this;
  218.     return a => {
  219.       var x = t.Run(a);
  220.       return x.Put(f(x.Pos));
  221.     };
  222.   }
  223.  
  224.   /*
  225.   Lenses compose.
  226.   */
  227.   public Lens<A, C> Then<C>(Lens<B, C> w) {
  228.     var t = this;
  229.     return Lens<A, C>.lens(a => {
  230.       Store<B, A> y = t.Run(a);
  231.       Store<C, B> z = w.Run(y.Pos);
  232.       return Store<C, A>.store(c => y.Put(z.Put(c)), z.Pos);
  233.     });
  234.   }
  235.  
  236.   /*
  237.   Lenses split on choice.
  238.   */
  239.   public Lens<Or<A, X>, B> Sum<X>(Lens<X, B> l) {
  240.     var t = this;
  241.     return Lens<Or<A, X>, B>.lens(e =>
  242.       e.Fold<Store<B, Or<A, X>>>(a => t.Run(a).Select(j => Or<A, X>.Left(j)), x => l.Run(x).Select(j => Or<A, X>.Right(j)))
  243.     );
  244.   }
  245.  
  246.   /*
  247.   Lenses are a tensor product.
  248.   */
  249.   public Lens<And<A, C>, And<B, D>> Product<C, D>(Lens<C, D> l) {
  250.     var t = this;
  251.     return Lens<And<A, C>, And<B, D>>.lens(v =>
  252.       t.Run(v.a).Product(l.Run(v.b)));
  253.   }
  254.  
  255.   public static Lens<A, B> lens(Func<A, Store<B, A>> f) {
  256.     return new Lens<A, B>(f);
  257.   }
  258.  
  259.   public static Lens<A, B> lens(Func<A, B, A> s, Func<A, B> g) {
  260.     return new Lens<A, B>(a =>
  261.       Store<B, A>.store(b => s(a, b), g(a)));
  262.   }
  263.  
  264.   /*
  265.   The lens identity for lens composition (Then method).
  266.   */
  267.   public static Lens<A, A> LensIdentity() {
  268.     return Lens<A, A>.lens(a => Store<A, A>.store(v => v, a));
  269.   }
  270.  
  271.   /*
  272.   A predicate is a lens.
  273.   */
  274.   public static Lens<Store<A, bool>, Or<A, A>> PredicateLens() {
  275.     return Lens<Store<A, bool>, Or<A, A>>.lens(s => {
  276.       var g = s.Pos;
  277.       return Store<Or<A, A>, Store<A, bool>>.store(o =>
  278.         o.Fold<Store<A, bool>>(l => And<bool, A>.and(true, l).ConstStore, r => And<bool, A>.and(false, r).ConstStore)
  279.       , s.Put(g) ? Or<A, A>.Left(g) : Or<A, A>.Right(g));
  280.     });
  281.   }
  282.  
  283.   /*
  284.   Lens unzips.
  285.   */
  286.   public static And<Lens<X, A>, Lens<X, B>> UnzipLens<X>(Lens<X, And<A, B>> l) {
  287.     return And<Lens<X, A>, Lens<X, B>>.and(
  288.       Lens<X, A>.lens(x => {
  289.         var c = l.Run(x);
  290.         var i = c.Pos;
  291.         return Store<A, X>.store(a => c.Put(And<A, B>.and(a, i.b)), i.a);        
  292.       })
  293.     , Lens<X, B>.lens(x => {
  294.         var c = l.Run(x);
  295.         var i = c.Pos;
  296.         return Store<B, X>.store(b => c.Put(And<A, B>.and(i.a, b)), i.b);
  297.       })
  298.     );
  299.   }
  300.  
  301.   /*
  302.   Lens factors.
  303.   */
  304.   public static Lens<Or<And<A, B>, And<A, C>>, And<A, Or<B, C>>> FactorLens<C>() {
  305.     return Lens<Or<And<A, B>, And<A, C>>, And<A, Or<B, C>>>.lens(e =>
  306.       Store<And<A, Or<B, C>>, Or<And<A, B>, And<A, C>>>.store(y =>
  307.         y.b.BinarySelect(b => And<A, B>.and(y.a, b), c => And<A, C>.and(y.a, c))
  308.       , e.Fold<And<A, Or<B, C>>>(
  309.           b => And<A, Or<B, C>>.and(b.a, Or<B, C>.Left(b.b))
  310.         , c => And<A, Or<B, C>>.and(c.a, Or<B, C>.Right(c.b)))));    
  311.   }
  312.  
  313.   /*
  314.   Lens distributes.
  315.   */
  316.   public static Lens<And<A, Or<B, C>>, Or<And<A, B>, And<A, C>>> DistributLens<C>() {
  317.     return Lens<And<A, Or<B, C>>, Or<And<A, B>, And<A, C>>>.lens(e =>
  318.       Store<Or<And<A, B>, And<A, C>>, And<A, Or<B, C>>>.store(y =>
  319.         y.Fold<And<A, Or<B, C>>>(
  320.           l => And<A, Or<B, C>>.and(l.a, Or<B, C>.Left(l.b))
  321.         , r => And<A, Or<B, C>>.and(r.a, Or<B, C>.Right(r.b)))
  322.       , e.b.BinarySelect(b => And<A, B>.and(e.a, b), c => And<A, C>.and(e.a, c))
  323.       ));
  324.   }
  325.  
  326.   /*
  327.   The lens for the first element of a pair.
  328.   */
  329.   public static Lens<And<A, B>, A> FirstLens() {
  330.     return Lens<And<A, B>, A>.lens(v =>
  331.       Store<A, And<A, B>>.store(a => And<A, B>.and(a, v.b), v.a));
  332.   }
  333.  
  334.   /*
  335.   The lens for the second element of a pair.
  336.   */
  337.   public static Lens<And<A, B>, B> SecondLens() {
  338.     return Lens<And<A, B>, B>.lens(v =>
  339.       Store<B, And<A, B>>.store(b => And<A, B>.and(v.a, b), v.b));
  340.   }
  341.  
  342.   /*
  343.   The three lens laws; identity, retention and double-set.
  344.  
  345.   All lenses must satisfy these laws.
  346.   i.e. It must not be possible to have these laws return false.
  347.   The two coalgebra laws given below follow from these three laws.
  348.   */
  349.   public bool IdentityLaw(A a) {
  350.     var c = Run(a);
  351.     return c.Put(c.Pos).Equals(a);
  352.   }
  353.  
  354.   public bool RetentionLaw(A a, B b) {
  355.     return Run(Run(a).Put(b)).Pos.Equals(b);
  356.   }
  357.  
  358.   public bool DoubleSetLaw(A a, B b1, B b2) {
  359.     var c = Run(a);
  360.     return Run(c.Put(b1)).Put(b2).Equals(c.Put(b2));
  361.   }
  362.  
  363.   /*
  364.   The coalgebra laws.
  365.  
  366.   All lenses must satisfy these laws.
  367.   i.e. It must not be possible to have these laws return false.
  368.   The three lens laws above follow from these two laws.
  369.   */
  370.   public bool CoalgebraLaw1(A a) {
  371.     return Run(a).CoPoint.Equals(a);
  372.   }
  373.  
  374.   public bool CoalgebraLaw2(A a) {
  375.     var t = this;
  376.     return Run(a).Select(r => t.Run(r)).Equals(t.Run(a).Duplicate);
  377.   }
  378.  
  379. }
  380.  
  381. }
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