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Set Universe Polymorphism.
Record Pred@{A} :=
{ car :> Type@{A}
; P : car -> Prop
}.
Class All@{A} (A : Pred@{A}) : Type :=
{ proof : forall (a : A), P A a
}.
Record Pred_All@{A} : Type :=
{ P' :> Pred@{A}
; P'_All : All P'
}.
Global Instance Pred_All_instance (A : Pred_All) : All A := P'_All A.
Definition Pred_All_proof {A : Pred_All} (a : A) : P A a.
Proof.
solve[auto using proof].
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