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(* Interaction between coercions and casts *)
(* Example provided by Eduardo Gimenez *)
Parameter Z S : Set.
Parameter f : S -> Z.
Coercion f : S >-> Z.
Parameter g : Z -> Z.
Check (fun s => g (s:S)).
(* Check uniform inheritance condition *)
Parameter h : nat -> nat -> Prop.
Parameter i : forall n m : nat, h n m -> nat.
Coercion i : h >-> nat.
(* Check coercion to funclass when the source occurs in the target *)
Parameter C : nat -> nat -> nat.
Coercion C : nat >-> Funclass.
(* Remark: in the following example, it cannot be decided whether C is
from nat to Funclass or from A to nat. An explicit Coercion command is
expected
Parameter A : nat -> Prop.
Parameter C:> forall n:nat, A n -> nat.
*)
(* Check coercion between products based on eta-expansion *)
(* (there was a de Bruijn bug until rev 9254) *)
Section P.
Variable E : Set.
Variables C D : E -> Prop.
Variable G :> forall x, C x -> D x.
Check fun (H : forall y:E, y = y -> C y) => (H : forall y:E, y = y -> D y).
End P.
(* Check that class arguments are computed the same when looking for a
coercion and when applying it (class_args_of) (failed until rev 9255) *)
Section Q.
Variable bool : Set.
Variables C D : bool -> Prop.
Variable G :> forall x, C x -> D x.
Variable f : nat -> bool.
Definition For_all (P : nat -> Prop) := forall x, P x.
Check fun (H : For_all (fun x => C (f x))) => H : forall x, D (f x).
Check fun (H : For_all (fun x => C (f x))) x => H x : D (f x).
Check fun (H : For_all (fun x => C (f x))) => H : For_all (fun x => D (f x)).
End Q.
(* Combining class lookup and path lookup so that if a lookup fails, another
descent in the class can be found (see wish #1934) *)
Record Setoid : Type :=
{ car :> Type }.
Record Morphism (X Y:Setoid) : Type :=
{evalMorphism :> X -> Y}.
Definition extSetoid (X Y:Setoid) : Setoid.
intros X Y.
constructor.
exact (Morphism X Y).
Defined.
Definition ClaimA := forall (X Y:Setoid) (f: extSetoid X Y) x, f x= f x.
Coercion irrelevent := (fun _ => I) : True -> car (Build_Setoid True).
Definition ClaimB := forall (X Y:Setoid) (f: extSetoid X Y) (x:X), f x= f x.
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