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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 := C1 (_:I1) (_:=0).
Check (fun x:I1 =>
match x with
| C1 i n => (i,n)
end).
(* Check implicit parameters of inductive types (submitted by Pierre
Casteran and also implicit in BZ#338) *)
Set Implicit Arguments.
Unset Strict Implicit.
CoInductive LList (A : Set) : Set :=
| LNil : LList A
| LCons : A -> LList A -> LList A.
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 BZ#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.
(* Check that test for binders capturing implicit arguments is not stronger
than needed (problem raised by Cedric Auger) *)
Set Implicit Arguments.
Inductive bool_comp2 (b: bool): bool -> Prop :=
| Opp2: forall q, (match b return Prop with
| true => match q return Prop with
true => False |
false => True end
| false => match q return Prop with
true => True |
false => False end end) -> bool_comp2 b q.
(* This one is still to be made acceptable...
Set Implicit Arguments.
Inductive I A : A->Prop := C a : (forall A, A) -> I a.
*)
(* Test recursively non-uniform parameters (was formerly in params_ind.v) *)
Inductive list (A : Set) : Set :=
| nil : list A
| cons : A -> list (A -> A) -> list A.
(* Check inference of evars in arity using information from constructors *)
Inductive foo1 : forall p, Prop := cc1 : foo1 0.
(* Check cross inference of evars from constructors *)
Inductive foo2 : forall p, Prop := cc2 : forall q, foo2 q | cc3 : foo2 0.
(* An example with reduction removing an occurrence of the inductive type in one of its argument *)
Inductive IND1 (A:Type) := CONS1 : IND1 ((fun x => A) IND1).
(* These types were considered as ill-formed before March 2015, while they
could be accepted considering that the type IND1 above was accepted *)
Inductive IND2 (A:Type) (T:=fun _ : Type->Type => A) := CONS2 : IND2 A -> IND2 (T IND2).
Inductive IND3 (A:Type) (T:=fun _ : Type->Type => A) := CONS3 : IND3 (T IND3) -> IND3 A.
Inductive IND4 (A:Type) := CONS4 : IND4 ((fun x => A) IND4) -> IND4 A.
(* This type was ok before March 2015 *)
Inductive IND5 (A : Type) (T := A) : Type := CONS5 : IND5 ((fun _ => A) 0) -> IND5 A.
(* An example of nested positivity which was rejected by the kernel
before 24 March 2015 (even with Unset Elimination Schemes to avoid
the _rect bug) due to the wrong computation of non-recursively
uniform parameters in list' *)
Inductive list' (A:Type) (B:=A) :=
| nil' : list' A
| cons' : A -> list' B -> list' A.
Inductive tree := node : list' tree -> tree.
(* This type was raising an anomaly when building the _rect scheme,
because of a bug in Inductiveops.get_arity in the presence of
let-ins and recursively non-uniform parameters. *)
Inductive L (A:Type) (T:=A) : Type := C : L nat -> L A.
(* This type was raising an anomaly when building the _rect scheme,
because of a wrong computation of the number of non-recursively
uniform parameters when conversion is needed, leading the example to
hit the Inductiveops.get_arity bug mentioned above (see #3491) *)
Inductive IND6 (A:Type) (T:=A) := CONS6 : IND6 T -> IND6 A.
Module TemplateProp.
(** Check lowering of a template universe polymorphic inductive to Prop *)
Inductive Foo (A : Type) : Type := foo : A -> Foo A.
Check Foo True : Prop.
End TemplateProp.
Module PolyNoLowerProp.
(** Check lowering of a general universe polymorphic inductive to Prop is _failing_ *)
Polymorphic Inductive Foo (A : Type) : Type := foo : A -> Foo A.
Fail Check Foo True : Prop.
End PolyNoLowerProp.
(* Test building of elimination scheme with noth let-ins and
non-recursively uniform parameters *)
Module NonRecLetIn.
Unset Implicit Arguments.
Inductive Ind (b:=2) (a:nat) (c:=1) : Type :=
| Base : Ind a
| Rec : Ind (S a) -> Ind a.
Check Ind_rect (fun n (b:Ind n) => b = b)
(fun n => eq_refl)
(fun n b c => f_equal (Rec n) eq_refl) 0 (Rec 0 (Base 1)).
End NonRecLetIn.
(* Test treatment of let-in in the definition of Records *)
(* Should fail with "Sort expected" *)
Fail Inductive foo (T : Type) : let T := Type in T :=
{ r : forall x : T, x = x }.
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