Definition MA_spec i P (R : Z -> mpred) Q := view_shift P (R i * (weak_precise_mpred (R i) && emp) * Q).Definition MA_type := ProdType (ProdType (ProdType (ConstType Z) Mpred) (ArrowType (ConstType Z) Mpred)) Mpred.Definition AL_spec P (R : Z -> mpred) Q := forall vx, repable_signed vx -> view_shift (R vx * P) (R vx * Q vx).Definition AL_type := ProdType (ProdType (ProdType (ConstType (share * val))
  Mpred) (ArrowType (ConstType Z) Mpred)) (ArrowType (ConstType Z) Mpred).Definition AS_spec v P (R : Z -> mpred) Q := forall vx, repable_signed vx ->
  view_shift (R vx * P)
  (R v * (weak_precise_mpred (R v) && emp) * Q).Definition AS_type := ProdType (ProdType (ProdType
  (ConstType (share * val * Z)) Mpred) (ArrowType (ConstType Z) Mpred)) Mpred.Definition ACAS_spec c v P (R Q : Z -> mpred) := forall vx, repable_signed vx ->
  view_shift (R vx * P)
  (R (if eq_dec c vx then v else vx) * (weak_precise_mpred (R (if eq_dec c vx then v else vx)) && emp) * Q vx).Definition ACAS_type := ProdType (ProdType (ProdType
  (ConstType (share * val * Z * Z)) Mpred)
  (ArrowType (ConstType Z) Mpred))
  (ArrowType (ConstType Z) Mpred).

Lemma A_inv_positive : forall x l R, positive_mpred (A_inv x l R).