chore: fib_correct monadic reasoning example as a test (#6647)

This PR records the `fib_impl n = fib_spec n` example, and a proof using
current technologies, as a test.

I'd like to think about eliminating `MProd` from the terms produced by
`do` notation; it seems (at least) a simproc would be required.

tests/lean/run/fib_correct.lean

@@ -0,0 +1,46 @@
+/-!
+This is an example for monadic reasoning.
+
+The eventual goal is to provide a nice user experience for proving `fib_impl n = fib_spec n`
+and related goals.
+
+Currently, this file just contains a proof that uses `simp` lemmas to convert the `do` notation
+and for loop into a `List.foldl`, and then gives a "functional" proof.
+(This is *not* the nice user experience we are aiming for!)
+
+Even in this setup, there is an awkward problem that `do` blocks handle multiple mutable variables
+via the universe monomorphic `MProd` type, to avoid universe unification issues arising when using
+`Prod`. We have to jump through some additional hoops to handle that.
+We could provide simp lemmas, simprocs, and possibly custom tactics to eliminator `MProd` from the
+terms produced by `do` notation.
+-/
+
+def fib_spec : Nat → Nat
+| 0 => 0
+| 1 => 1
+| n+2 => fib_spec n + fib_spec (n+1)
+
+def fib_impl (n : Nat) := Id.run do
+  if n = 0 then return 0
+  let mut a := 0
+  let mut b := 0
+  b := b + 1
+  for _ in [1:n] do
+    let a' := a
+    a := b
+    b := a' + b
+  return b
+
+theorem fib_correct {n} : fib_impl n = fib_spec n := by
+  -- The default simp set eliminates the binds generated by `do` notation,
+  -- and converts the `for` loop into a `List.foldl` over `List.range'`.
+  simp [fib_impl, Id.run]
+  match n with
+  | 0 => simp [fib_spec]
+  | n+1 =>
+    -- Note here that we have to use `⟨x, y⟩ : MProd _ _`, because these are not `Prod` products.
+    suffices ((List.range' 1 n).foldl (fun b a ↦ ⟨b.snd, b.fst + b.snd⟩) (⟨0, 1⟩ : MProd _ _)) =
+        ⟨fib_spec n, fib_spec (n + 1)⟩ by simp_all
+    induction n with
+    | zero => rfl
+    | succ n ih => simp [fib_spec, List.range'_1_concat, ih]