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(* Test des definitions inductives imbriquees *)
Require Import List.
Inductive X : Set :=
cons1 : list X -> X.
Inductive Y : Set :=
cons2 : list (Y * Y) -> Y.
(* Test inductive types with local definitions (arity) *)
Inductive eq1 : forall A:Type, let B:=A in A -> Prop :=
refl1 : eq1 True I.
Check
fun (P : forall A : Type, let B := A in A -> Type) (f : P True I) (A : Type) =>
let B := A in
fun (a : A) (e : eq1 A a) =>
match e in (eq1 A0 B0 a0) return (P A0 a0) with
| refl1 => f
end.
Inductive eq2 (A:Type) (a:A)
: forall B C:Type, let D:=(A*B*C)%type in D -> Prop :=
refl2 : eq2 A a unit bool (a,tt,true).
(* Check inductive types with local definitions (parameters) *)
Inductive A (C D : Prop) (E:=C) (F:=D) (x y : E -> F) : E -> Set :=
I : forall z : E, A C D x y z.
Check
(fun C D : Prop =>
let E := C in
let F := D in
fun (x y : E -> F) (P : forall c : C, A C D x y c -> Type)
(f : forall z : C, P z (I C D x y z)) (y0 : C)
(a : A C D x y y0) =>
match a as a0 in (A _ _ _ _ _ _ y1) return (P y1 a0) with
| I x0 => f x0
end).
Record B (C D : Set) (E:=C) (F:=D) (x y : E -> F) : Set := {p : C; q : E}.
Check
(fun C D : Set =>
let E := C in
let F := D in
fun (x y : E -> F) (P : B C D x y -> Type)
(f : forall p0 q0 : C, P (Build_B C D x y p0 q0))
(b : B C D x y) =>
match b as b0 return (P b0) with
| Build_B x0 x1 => f x0 x1
end).
(* Check inductive types with local definitions (constructors) *)
Inductive I1 : Set := c (_:I1) (_:=0).
Check (fun x:I1 =>
match x with
| c i n => (i,n)
end).
(* Check implicit parameters of inductive types (submitted by Pierre
Casteran and also implicit in #338) *)
Set Implicit Arguments.
Unset Strict Implicit.
CoInductive LList (A : Set) : Set :=
| LNil : LList A
| LCons : A -> LList A -> LList A.
Implicit Arguments LNil [A].
Inductive Finite (A : Set) : LList A -> Prop :=
| Finite_LNil : Finite LNil
| Finite_LCons :
forall (a : A) (l : LList A), Finite l -> Finite (LCons a l).
(* Check positivity modulo reduction (cf bug #983) *)
Record P:Type := {PA:Set; PB:Set}.
Definition F (p:P) := (PA p) -> (PB p).
Inductive I_F:Set := c : (F (Build_P nat I_F)) -> I_F.
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