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+(* Check pattern-unification on evars in apply unification *)
+
+Axiom app : forall tau tau':Type, (tau -> tau') -> tau -> tau'.
+
+Axiom copy : forall tau:Type, tau -> tau -> Prop.
+Axiom copyr : forall tau:Type, tau -> tau -> Prop.
+Axiom copyf : forall tau:Type, tau -> tau -> Prop.
+Axiom eq : forall tau:Type, tau -> tau -> Prop.
+Axiom subst : forall tau tau':Type, (tau -> tau') -> tau -> tau' -> Prop.
+
+Axiom copy_atom : forall tau:Type, forall t t':tau, eq tau t t' -> copy tau t t'.
+Axiom copy_fun: forall tau tau':Type, forall t t':(tau->tau'),
+(forall x:tau, copyr tau x x->copy tau' (t x) (t' x))
+->copy (tau->tau') t t'.
+
+Axiom copyr_atom : forall tau:Type, forall t t':tau, copyr tau t t' -> eq tau t t'.
+Axiom copyr_fun: forall tau tau':Type, forall t t':(tau->tau'),
+copyr (tau->tau') t t'
+->(forall x y:tau, copy tau x y->copyr tau' (t x) (t' y)).
+
+Axiom copyf_atom : forall tau:Type, forall t t':tau, copyf tau t t' -> eq tau t t'.
+Axiom copyf_fun: forall tau tau':Type, forall t t':(tau->tau'),
+copyr (tau->tau') t t'
+->(forall x y:tau, forall z1 z2:tau',
+(copy tau x y)->
+(subst tau tau' t x z1)->
+(subst tau tau' t' y z2)->
+copyf tau' z1 z2).
+
+Axiom eqappg: forall tau tau':Type, forall t:tau->tau', forall q:tau, forall r:tau',forall t':tau',
+( ((subst tau tau' t q t') /\ (eq tau' t' r))
+->eq tau' (app tau tau' t q) r).
+
+Axiom eqappd: forall tau tau':Type, forall t:tau->tau', forall q:tau, forall r:tau',
+forall t':tau', ((subst tau tau' t q t') /\ (eq tau' r t'))
+->eq tau' r (app tau tau' t q).
+
+Axiom substcopy: forall tau tau':Type, forall t:tau->tau', forall q:tau, forall r:tau',
+(forall x:tau, (copyf tau x q) -> (copy tau' (t x) r))
+->subst tau tau' t q r.
+
+Ltac EtaLong := (apply copy_fun;intros;EtaLong)|| apply copy_atom.
+Ltac Subst := apply substcopy;intros;EtaLong.
+Ltac Rigid_aux := fun A => apply A|| Rigid_aux (copyr_fun _ _ _ _ A).
+Ltac Rigid := fun A => apply copyr_atom; Rigid_aux A.
+
+Theorem church0: forall i:Type, exists X:(i->i)->i->i,
+copy ((i->i)->i->i) (fun f:i->i => fun x:i=>f (X f x)) (fun f:i->i=>fun x:i=>app i i (X f) (f x)).
+intros.
+esplit.
+EtaLong.
+eapply eqappd;split.
+Subst.
+apply copyf_atom.
+Show Existentials.
+apply H1.