Fix RHole, add test files, papers, piforall examples

piforall/Fix.pi

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+-- Can we define the Y combinator in pi-forall?
+-- Yes! See below.
+-- Note: pi-forall allows recursive definitions,
+-- so this is not necessary at all.
+
+module Fix where
+
+-- To type check the Y combinator, we need to have a type
+-- D such that D ~~ D -> D
+
+
+data D (A : Type) : Type where
+   F of (_ : D A -> D A)
+   V of (_ : A)
+
+unV : [A:Type] -> D A -> A
+unV = \[A] v.
+        case v of
+          V y -> y
+          F f -> TRUSTME
+
+unF :[A:Type] -> D A -> D A -> D A
+unF = \[A] v x .
+   case v of
+     F f -> f x
+     V y -> TRUSTME
+
+-- Here's the Y-combinator. To make it type
+-- check, we need to add the appropriate conversions
+-- into and out of the D type.
+
+fix : [A:Type] -> (A -> A) -> A
+fix = \ [A] g.
+   let omega =
+      ( \x. V (g (unV [A] (unF [A] x x)))
+      : D A -> D A) in
+      unV [A] (omega (F omega))
+
+-- Example use case
+
+
+data Nat : Type where
+   Zero
+   Succ of ( _ : Nat)
+
+fix_add : Nat -> Nat -> Nat
+fix_add = fix [Nat -> Nat -> Nat]
+        \radd. \x. \y.
+            case x of
+              Zero   -> y
+              Succ n -> Succ (radd n y)
+
+test : fix_add 5 2 = 7
+test = Refl