2 files changed, 102 insertions, 0 deletions
diff --git a/Test/dafny4/MonadicLaws.dfy b/Test/dafny4/MonadicLaws.dfy
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+++ b/Test/dafny4/MonadicLaws.dfy
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+// RUN: %dafny /compile:0 /rprint:"%t.rprint" /autoTriggers:1 "%s" > "%t"
+// RUN: %diff "%s.expect" "%t"
+
+// Monadic Laws
+// Niki Vazou and Rustan Leino
+// 28 March 2016
+
+datatype List<T> = Nil | Cons(head: T, tail: List)
+
+predicate Total<T,U>(p: T -> U)
+  reads p.reads
+{
+  forall x :: p.reads(x) == {} && p.requires(x)
+}
+
+function append(xs: List, ys: List): List
+{
+  match xs
+  case Nil => ys
+  case Cons(x, xs') => Cons(x, append(xs', ys))
+}
+
+lemma AppendNil(xs: List)
+  ensures append(xs, Nil) == xs
+{
+}
+
+lemma AppendAssoc(xs: List, ys: List, zs: List)
+  ensures append(append(xs, ys), zs) == append(xs, append(ys, zs));
+{
+}
+
+function Return<T>(a: T): List
+{
+  Cons(a, Nil)
+}
+
+function Bind<T,U>(xs: List<T>, f: T -> List<U>): List<U>
+  requires Total(f)
+{
+  match xs
+  case Nil => Nil
+  case Cons(x, xs') => append(f(x), Bind(xs', f))
+}
+
+lemma LeftIdentity<T>(a: T, f: T -> List)
+  requires Total(f)
+  ensures Bind(Return(a), f) == f(a)
+{
+  AppendNil(f(a));
+}
+
+lemma RightIdentity<T>(m: List)
+  ensures Bind(m, Return) == m
+{
+  match m
+  case Nil =>
+    assert Bind<T,T>(Nil, Return) == Nil;
+  case Cons(x, m') =>
+    calc {
+      Bind(Cons(x, m'), Return);
+      append(Return(x), Bind(m', Return));
+      Cons(x, Bind(m', Return));
+    }
+}
+
+lemma Associativity<T>(m: List, f: T -> List, g: T -> List)
+  requires Total(f) && Total(g)
+  ensures Bind(Bind(m, f), g) == Bind(m, x => Bind(f(x), g))
+{
+  match m
+  case Nil =>
+    assert Bind(m, x => Bind(f(x), g)) == Nil;
+  case Cons(x, xs) =>
+    match f(x)
+    case Nil =>
+      calc {
+        Bind(xs, y => Bind(f(y), g));
+        Bind(Cons(x, xs), y => Bind(f(y), g));
+      }
+    case Cons(y, ys) =>
+      calc {
+        append(g(y), Bind(append(ys, Bind(xs, f)), g));
+        { BindOverAppend(ys, Bind(xs, f), g); }
+        append(g(y), append(Bind(ys, g), Bind(Bind(xs, f), g)));
+        { AppendAssoc(g(y), Bind(ys, g), Bind(Bind(xs, f), g)); }
+        append(append(g(y), Bind(ys, g)), Bind(Bind(xs, f), g));
+        Bind(Cons(x, xs), z => Bind(f(z), g));
+      }
+}
+
+lemma BindOverAppend<T>(xs: List, ys: List, g: T -> List)
+  requires Total(g)
+  ensures Bind(append(xs, ys), g) == append(Bind(xs, g), Bind(ys, g))
+{
+  match xs
+  case Nil =>
+  case Cons(x, xs') =>
+    AppendAssoc(g(x), Bind(xs', g), Bind(ys, g));
+}
diff --git a/Test/dafny4/MonadicLaws.dfy.expect b/Test/dafny4/MonadicLaws.dfy.expect
new file mode 100644
index 00000000..d903c7c5
--- /dev/null
+++ b/Test/dafny4/MonadicLaws.dfy.expect
@@ -0,0 +1,2 @@
+
+Dafny program verifier finished with 16 verified, 0 errors