example of inductive definitions: transitive closure

tutorials/inductive-witnesses/TransitiveClosure.scala

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+import stainless.annotation.*
+import stainless.lang.*
+
+/* This example shows how to represent inductive definitions in
+   Stainless using algebraic data types with invariants and explicit
+   proof witnesses. This is in the style of Coq. It is often not
+   needed in Stainless, but can be useful for complex proofs.
+ */
+
+object TransitiveClosure:
+  /* R is an example relation. @opaque means it is not expanded. */
+  @opaque
+  def R(x: BigInt, y: BigInt): Boolean = 
+    y == x + 2
+
+  /* We define proof witnesses for reflexive transitive closure R*(x,y).
+     We can think of this as an inductive definition. */
+
+  /* I would like ClosureProof to have two parameters, but Stainless does not support this yet. 
+     Instead, we define two getters. */
+
+  sealed trait ClosureProof:
+    def x: BigInt
+    def y: BigInt
+
+  case class Equal(v: BigInt) extends ClosureProof:
+    override def x: BigInt = v
+    override def y: BigInt = v
+
+  case class ByBaseR(v1: BigInt, v2: BigInt) extends ClosureProof:
+    require(R(v1, v2))
+    override def x: BigInt = v1
+    override def y: BigInt = v2
+
+  case class ByTransitivity(p1: ClosureProof, p2: ClosureProof) extends ClosureProof:
+    require(p1.y == p2.x)
+    override def x: BigInt = p1.x
+    override def y: BigInt = p2.y
+
+  def Rstar(using proof: ClosureProof)(x: BigInt, y: BigInt): Boolean = 
+    proof.x == x && proof.y == y
+
+  /* Let us prove that Rstar(x,y) implies y - x is even and non-negative. */
+  /* Proof of that fact is induction on the proof tree for R*(x,y).  */
+
+  def soundnessRstar(using proof: ClosureProof)(x: BigInt, y: BigInt): Unit = {
+    require(Rstar(x, y))
+    proof match
+      case Equal(v) => ()
+      case ByBaseR(v1, v2) => unfold(R(v1, v2))
+      case ByTransitivity(p1, p2) => 
+        soundnessRstar(using p1)(p1.x, p1.y)
+        soundnessRstar(using p2)(p2.x, p2.y)
+  }.ensuring(_ => x <= y && (y - x) % 2 == 0)
+
+  /* Let us now prove the converse: if y - x is even and non-negative, then Rstar(x,y). */
+  /* Proof is induction on y-x. */
+
+  def completenessRstar(x: BigInt, y: BigInt): ClosureProof = {
+    require(x <= y && (y - x) % 2 == 0)
+    decreases(y - x)
+    if y - x == 0 then Equal(x)
+    else if y - x == 2 then 
+      unfold(R(x, y))
+      ByBaseR(x, y)
+    else
+      val p1 = completenessRstar(x, x + 2)
+      val p2 = completenessRstar(x + 2, y)
+      ByTransitivity(p1, p2)
+
+  }.ensuring(proof => proof.x == x && proof.y == y)
+
+
+end TransitiveClosure