Set Implicit Arguments.
From Undecidability.HOU Require Import std.decidable.
Inductive diag: nat -> nat -> Type :=
| diagB: diag 0 0
| diagC: forall m, diag 0 m -> diag (S m) 0
| diagS: forall n m, diag (S n) m -> diag n (S m).
Fixpoint diagStepR m n (d: diag m n) : diag m (S n) :=
match d with
| diagB => diagS (diagC diagB)
| diagC H => diagS (diagC (diagStepR H))
| diagS H => diagS (diagStepR H)
end.
Fixpoint diagStepL m n (d: diag m n) : diag (S m) n :=
match d with
| diagB => diagC diagB
| diagC H => diagC (diagS (diagStepL H))
| diagS H => diagS (diagStepL H)
end.
Fixpoint diagZero (a: nat) : diag 0 a :=
match a with
| 0 => diagB
| S a => diagStepR (diagZero a)
end.
Fixpoint diagId m n: diag m n :=
match m with
| O => diagZero n
| S m => diagStepL (diagId m n)
end.
Definition I__P (p: nat * nat) :=
let (m, n) := p in
diag_rect (fun _ _ _ => nat) 0 (fun _ _ => S)  (fun _ _ _ => S) (diagId m n): nat.
Fixpoint R__P (n: nat) : nat * nat  :=
match n with
| 0 =>  (0,0)
| S n => match R__P n with
| (0, b)  => (S b, 0)
| (S a, b) => (a, S b)
end
end.
Definition I__S (s: nat + nat) :=
match s with
| inl n => double n
| inr n => S (double n)
end.
Definition R__S (n: nat) :=
if even n then inl (div2 n) else inr (div2 n).

Lemma I__S_R__S n: I__S (R__S n) = n.