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Set Universe Polymorphism.
Set Polymorphic Inductive Cumulativity.
Set Printing Universes.
Module Simple.
(* in the real world foo@{i} might be [@paths@{i} nat] I guess *)
Inductive foo : nat -> Type :=.
(* on [refl (fun x => f x)] this computes to [refl f] *)
Definition eta_out {A B} (f g : forall x : A, B x) (e : (fun x => f x) = (fun x => g x)) : f = g.
Proof.
change (f = g) in e. destruct e;reflexivity.
Defined.
Section univs.
Universes i j.
Constraint i < j. (* fail instead of forcing equality *)
Definition one : (fun n => foo@{i} n) = fun n => foo@{j} n := eq_refl.
Definition two : foo@{i} = foo@{j} := eta_out _ _ one.
Definition two' : foo@{i} = foo@{j} := Eval compute in two.
Definition three := @eq_refl (foo@{i} = foo@{j}) two.
Definition four := Eval compute in three.
Definition five : foo@{i} = foo@{j} := eq_refl.
End univs.
(* inference tries and succeeds with syntactic equality which doesn't eta expand *)
Fail Definition infer@{i j k|i < k, j < k, k < eq.u0}
: foo@{i} = foo@{j} :> (nat -> Type@{k})
:= eq_refl.
End Simple.
Module WithRed.
(** this test needs to reduce the parameter's type to work *)
Inductive foo@{i j} (b:bool) (x:if b return Type@{j} then Type@{i} else nat) : Type@{i} := .
(* on [refl (fun x => f x)] this computes to [refl f] *)
Definition eta_out {A B} (f g : forall x : A, B x) (e : (fun x => f x) = (fun x => g x)) : f = g.
Proof.
change (f = g) in e. destruct e;reflexivity.
Defined.
Section univs.
Universes i j k.
Constraint i < j. (* fail instead of forcing equality *)
Definition one : (fun n => foo@{i k} false n) = fun n => foo@{j k} false n := eq_refl.
Definition two : foo@{i k} false = foo@{j k} false := eta_out _ _ one.
Definition two' : foo@{i k} false = foo@{j k} false := Eval compute in two.
(* Failure of SR doesn't just mean that the type changes, sometimes
we lose being well-typed entirely. *)
Definition three := @eq_refl (foo@{i k} false = foo@{j k} false) two.
Definition four := Eval compute in three.
Definition five : foo@{i k} false = foo@{j k} false := eq_refl.
End univs.
(* inference tries and succeeds with syntactic equality which doesn't eta expand *)
Fail Definition infer@{i j k|i < k, j < k, k < eq.u0}
: foo@{i k} false = foo@{j k} false :> (nat -> Type@{k})
:= eq_refl.
End WithRed.
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