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(* This example was proposed by Cuihtlauac ALVARADO *)
Require Import List.
Fixpoint mult2 (n : nat) : nat :=
match n with
| O => 0
| S n => S (S (mult2 n))
end.
Inductive list : nat -> Set :=
| nil : list 0
| cons : forall n : nat, list (mult2 n) -> list (S (S (mult2 n))).
Type
(fun (P : forall n : nat, list n -> Prop) (f : P 0 nil)
(f0 : forall (n : nat) (l : list (mult2 n)),
P (mult2 n) l -> P (S (S (mult2 n))) (cons n l)) =>
fix F (n : nat) (l : list n) {struct l} : P n l :=
match l as x0 in (list x) return (P x x0) with
| nil => f
| cons n0 l0 => f0 n0 l0 (F (mult2 n0) l0)
end).
Inductive list' : nat -> Set :=
| nil' : list' 0
| cons' : forall n : nat, let m := mult2 n in list' m -> list' (S (S m)).
Fixpoint length n (l : list' n) {struct l} : nat :=
match l with
| nil' => 0
| cons' _ m l0 => S (length m l0)
end.
Type
(fun (P : forall n : nat, list' n -> Prop) (f : P 0 nil')
(f0 : forall n : nat,
let m := mult2 n in
forall l : list' m, P m l -> P (S (S m)) (cons' n l)) =>
fix F (n : nat) (l : list' n) {struct l} : P n l :=
match l as x0 in (list' x) return (P x x0) with
| nil' => f
| cons' n0 m l0 => f0 n0 l0 (F m l0)
end).
(* Check on-the-fly insertion of let-in patterns for compatibility *)
Inductive list'' : nat -> Set :=
| nil'' : list'' 0
| cons'' :
forall n : nat,
let m := mult2 n in list'' m -> let p := S (S m) in list'' p.
Check
(fix length n (l : list'' n) {struct l} : nat :=
match l with
| nil'' => 0
| cons'' n l0 => S (length (mult2 n) l0)
end).
(* Check let-in in both parameters and in constructors *)
Inductive list''' (A:Set) (B:=(A*A)%type) (a:A) : B -> Set :=
| nil''' : list''' A a (a,a)
| cons''' :
forall a' : A, let m := (a',a) in list''' A a m -> list''' A a (a,a).
Fixpoint length''' (A:Set) (B:=(A*A)%type) (a:A) (m:B) (l:list''' A a m)
{struct l} : nat :=
match l with
| nil''' => 0
| cons''' _ m l0 => S (length''' A a m l0)
end.
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