Ltac uniq_assert name P := 
 lazymatch goal with H: P |- _ => fail 
    | _ => let H1 := fresh "H" name in assert (H1:P) end.Ltac readable_unreadable_join_prover := 
repeat match goal with
| H: join ?A ?B ?C, H1: ~readable_share ?C |- _ =>
   uniq_assert A (~readable_share A);
    [ clear - H H1; contradict H1; eapply join_readable1; eauto; fail | ]
| H: join ?A ?B ?C, H1: ~readable_share ?C |- _ =>
   uniq_assert B (~readable_share B);
    [ clear - H H1; contradict H1; eapply join_readable2; eauto; fail | ]
| H: join ?A ?B ?C, H0: ~readable_share ?B, H1: readable_share ?C |- _ =>
    (uniq_assert A (readable_share A);
    [ clear - H H0 H1; destruct (readable_share_dec A); 
      [solve [auto]
       |eapply join_unreadable_shares in H; eauto; solve [contradiction]] | ])
| H: join ?A ?B ?C, H0: ~readable_share ?A, H1: readable_share ?C |- _ =>
    (uniq_assert B (readable_share B);
    [ clear - H H0 H1; destruct (readable_share_dec B); 
      [solve [auto]
       | apply join_comm in H; 
         eapply join_unreadable_shares in H; eauto; solve [contradiction]] | ])
end.(*Lemma Cross_resource: Cross_alg resource.
Proof.
intro; intros.
destruct a as [ra | ra sa ka pa | ka pa | ma].
destruct b as [rb | rb sb kb pb | kb pb |]; try solve [elimtype False; inv H].
destruct z as [rz | rz sz kz pz | kz pz |]; try solve [elimtype False; inv H].
destruct c as [rc | rc sc kc pc | kc pc |]; try solve [elimtype False; inv H0].
destruct d as [rd | rd sd kd pd | kd pd |]; try solve [elimtype False; inv H0].
assert (J1: join ra rb rz) by (inv H; auto).
assert (J2: join rc rd rz) by (inv H0; auto).
destruct (share_cross_split _ _ _ _ _ J1 J2) as [[[[ac ad] bc] bd] [Ha [Hb [Hc Hd]]]].
readable_unreadable_join_prover.
exists (NO ac Hac,NO ad Had, NO bc Hbc, NO bd Hbd); 
  repeat split; simpl; auto; constructor; auto.
destruct z as [rz | rz sz kz pz | kz pz |]; try solve [elimtype False; inv H].
destruct c as [rc | rc sc kc pc | kc pc |]; try solve [elimtype False; inv H0].
destruct d as [rd | rd sd kd pd | kd pd |]; try solve [elimtype False; inv H0].
assert (J1: join ra rb rz) by (inv H; auto).
assert (J2: join rc rd rz) by (inv H0; auto).
destruct (share_cross_split _ _ _ _ _ J1 J2) as [[[[ac ad] bc] bd] [Ha [Hb [Hc Hd]]]].
readable_unreadable_join_prover.
exists (NO ac Hac, NO ad Had, NO bc Hbc, YES bd Hbd kb pb); inv H; inv H0;
  repeat split; simpl; auto; try constructor; auto.
assert (J1: join ra rb rz) by (inv H; auto).
destruct d as [rd | rd sd kd pd | kd pd |]; try solve [elimtype False; inv H0].
assert (J2: join rc rd rz) by (inv H0; auto).
destruct (share_cross_split _ _ _ _ _ J1 J2) as [[[[ac ad] bc] bd] [Ha [Hb [Hc Hd]]]].
readable_unreadable_join_prover.
exists (NO ac Hac, NO ad Had, YES bc Hbc kb pb, NO bd Hbd); inv H; inv H0;
  repeat split; simpl; auto; try constructor; auto.
assert (J2: join rc rd rz) by (inv H0; auto).
destruct (share_cross_split _ _ _ _ _ J1 J2) as [[[[ac ad] bc] bd] [Ha [Hb [Hc Hd]]]].
readable_unreadable_join_prover.
exists (NO ac Hac, NO ad Had, YES bc Hbc kb pb, YES bd Hbd kd pd); inv H; inv H0;
  repeat split; simpl; auto; try constructor; auto.
destruct b as [rb | rb sb kb pb | kb pb |]; try solve [elimtype False; inv H].
destruct z as [rz | rz sz kz pz | kz pz |]; try solve [elimtype False; inv H].
assert (J1: join ra rb rz) by (inv H; auto).
destruct c as [rc | rc sc kc pc | kc pc |]; try solve [elimtype False; inv H0].
destruct d as [rd | rd sd kd pd | kd pd |]; try solve [elimtype False; inv H0].
assert (J2: join rc rd rz) by (inv H0; auto).
destruct (share_cross_split _ _ _ _ _ J1 J2) as [[[[ac ad] bc] bd] [Ha [Hb [Hc Hd]]]].
readable_unreadable_join_prover.
exists (NO ac Hac, YES ad Had kd pd, NO bc Hbc, NO bd Hbd); inv H; inv H0;
  repeat split; simpl; auto; try constructor; auto.
destruct d as [rd | rd sd kd pd | kd pd |]; try solve [elimtype False; inv H0].
assert (J2: join rc rd rz) by (inv H0; auto).
destruct (share_cross_split _ _ _ _ _ J1 J2) as [[[[ac ad] bc] bd] [Ha [Hb [Hc Hd]]]].
readable_unreadable_join_prover.
exists (YES ac Hac kc pc, NO ad Had, NO bc Hbc, NO bd Hbd); inv H; inv H0;
  repeat split; simpl; auto; try constructor; auto.
assert (J2: join rc rd rz) by (inv H0; auto).
destruct (share_cross_split _ _ _ _ _ J1 J2) as [[[[ac ad] bc] bd] [Ha [Hb [Hc Hd]]]].
readable_unreadable_join_prover.
exists (YES ac Hac kc pc, YES ad Had kd pd, NO bc Hbc, NO bd Hbd); inv H; inv H0;
  repeat split; simpl; auto; try constructor; auto.
destruct z as [rz | rz sz kz pz | kz pz |]; try solve [elimtype False; inv H].
assert (J1: join ra rb rz) by (inv H; auto).
destruct c as [rc | rc sc kc pc | kc pc |]; try solve [elimtype False; inv H0].
destruct d as [rd | rd sd kd pd | kd pd |]; try solve [elimtype False; inv H0].
assert (J2: join rc rd rz) by (inv H0; auto).
destruct (share_cross_split _ _ _ _ _ J1 J2) as [[[[ac ad] bc] bd] [Ha [Hb [Hc Hd]]]].
readable_unreadable_join_prover.
exists (NO ac Hac, YES ad Had kd pd, NO bc Hbc, YES bd Hbd kd pd); inv H; inv H0;
  repeat split; simpl; auto; try constructor; auto.
destruct d as [rd | rd sd kd pd | kd pd |]; try solve [elimtype False; inv H0].
assert (J2: join rc rd rz) by (inv H0; auto).
destruct (share_cross_split _ _ _ _ _ J1 J2) as [[[[ac ad] bc] bd] [Ha [Hb [Hc Hd]]]].
readable_unreadable_join_prover.
exists (YES ac Hac kc pc, NO ad Had, YES bc Hbc kb pb, NO bd Hbd); inv H; inv H0;
  repeat split; simpl; auto; try constructor; auto.
assert (J2: join rc rd rz) by (inv H0; auto).
destruct (share_cross_split _ _ _ _ _ J1 J2) as [[[[ac ad] bc] bd] [Ha [Hb [Hc Hd]]]].
destruct (Sumbool.sumbool_not _ _ (readable_share_dec ac)) as [Hac|Hac].
readable_unreadable_join_prover.
destruct (Sumbool.sumbool_not _ _ (readable_share_dec bd)) as [Hbd|Hbd].
exists (NO ac Hac, YES ad Had ka pa, YES bc Hbc kc pc, NO bd Hbd); 
   inv H; inv H0; simpl; repeat split; auto;  constructor; auto.
exists (NO ac Hac, YES ad Had ka pa, YES bc Hbc kc pc, YES bd Hbd kd pd); 
   inv H; inv H0; simpl; repeat split; auto;  constructor; auto.
destruct (Sumbool.sumbool_not _ _ (readable_share_dec ad)) as [Had|Had];
readable_unreadable_join_prover;
destruct (Sumbool.sumbool_not _ _ (readable_share_dec bc)) as [Hbc|Hbc];
readable_unreadable_join_prover.
exists (YES ac Hac ka pa, NO ad Had, NO bc Hbc, YES bd Hbd kb pb); 
   inv H; inv H0; simpl; repeat split; auto;  constructor; auto.
exists (YES ac Hac ka pa, NO ad Had, YES bc Hbc kc pc, YES bd Hbd kd pd);
    inv H; inv H0; simpl; repeat split; auto;  constructor; auto.
exists (YES ac Hac kc pc, YES ad Had kc pc, NO bc Hbc, YES bd Hbd kb pb);
    inv H; inv H0; simpl; repeat split; auto;  constructor; auto.
destruct (Sumbool.sumbool_not _ _ (readable_share_dec bd)) as [Hbd|Hbd].
exists (YES ac Hac ka pa,  YES ad Had kd pd, YES bc Hbc kb pb, NO bd Hbd);
    inv H; inv H0; simpl; repeat split; auto;  constructor; auto.
exists (YES ac Hac ka pa, YES ad Had ka pa, 
       YES bc Hbc ka pa,  YES bd Hbd ka pa);
    inv H; inv H0; simpl; repeat split; auto;  constructor; auto.
exists (PURE ka pa, PURE ka pa, PURE ka pa, PURE ka pa).
inv H. inv H0.
repeat split; constructor; auto.
destruct b as [| | | mb]; try solve [elimtype False; inv H].
destruct z as [| | | mz]; try solve [elimtype False; inv H].
destruct c as [| | | mc]; try solve [elimtype False; inv H0].
destruct d as [| | | md]; try solve [elimtype False; inv H0].
(* relies on cross-split for ghost state *)
Qed.*)

Definition res_retain (r: resource) : Share.t :=
 match r with
  | NO sh _ => retainer_part sh
  | YES sh _ _ _ => retainer_part sh
  | PURE _ _ => Share.bot
 end.

Lemma core_NO: forall sh nsh, core (NO sh nsh) = NO Share.bot bot_unreadable.