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Set Implicit Arguments.
Require Import Logic.
Set Asymmetric Patterns.
Set Record Elimination Schemes.
Set Primitive Projections.
Record prod (A B : Type) : Type :=
pair { fst : A; snd : B }.
Print prod_rect.
(** prod_rect =
fun (A B : Type) (P : prod A B -> Type)
(f : forall (fst : A) (snd : B), P {| fst := fst; snd := snd |})
(p : prod A B) =>
match p as p0 return (P p0) with
| {| fst := x; snd := x0 |} => f x x0
end
: forall (A B : Type) (P : prod A B -> Type),
(forall (fst : A) (snd : B), P {| fst := fst; snd := snd |}) ->
forall p : prod A B, P p
Arguments A, B are implicit
Argument scopes are [type_scope type_scope _ _ _]
*)
(* What I really want: *)
Definition prod_rect' A B (P : prod A B -> Type) (u : forall (fst : A) (snd : B), P (pair fst snd))
(p : prod A B) : P p
:= u (fst p) (snd p).
Notation typeof x := (ltac:(let T := type of x in exact T)) (only parsing).
(* Check for eta *)
Check eq_refl : typeof (@prod_rect) = typeof (@prod_rect').
(* Check for the recursion principle I want *)
Check eq_refl : @prod_rect = @prod_rect'.
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