Definition int_type_min_max (type_i type_hi: type): option (Z * Z) :=
  match type_i, type_hi with
  | Tint I32 s_i _, Tint i_hi s_hi _ =>
    Some match s_i with
         | Signed => (match i_hi, s_hi with
                      | I32, Unsigned => 0
                      | _, _ => Int.min_signed
                      end, Int.max_signed)
         | Unsigned => (0, Int.max_unsigned)
         end
  | Tint I32 s_i _, Tlong s_hi _ =>
    Some match s_i with
         | Signed => match s_hi with
                     | Unsigned => (0, Int.max_signed)
                     | _ => (Int.min_signed, Int.max_signed)
                     end
         | Unsigned => (0, Int.max_unsigned)
         end
  | _, _ => None
  end.Inductive range_init_hl (type_lo type_i type_hi: type): (Z -> Z -> Prop) -> Prop :=
| construct_range_init_hl:
  forall int_min int_max,
    is_int_type type_lo = true ->
    int_type_min_max type_i type_hi = Some (int_min, int_max) ->
    range_init_hl type_lo type_i type_hi (fun m n => int_min <= m <= n /\ n <= int_max).Inductive range_init_hl (type_lo type_i type_hi: type): (Z -> Z -> Prop) -> Prop :=
| construct_range_init_hl:
  forall int_min int_max,
    is_int_type type_lo = true ->
    int_type_min_max type_i type_hi = Some (int_min, int_max) ->
    range_init_hl type_lo type_i type_hi (fun m n => int_min <= m <= n /\ n <= int_max).(*
Lemma range_init_hl_spec: forall (type_lo type_i type_hi: type) range,
  range_init_hl type_lo type_i type_hi range ->
  exists int_min_lo int_max_lo int_min_i int_max_i int_min_hi int_max_hi
    int_min int_max,
    is_int32_type type_lo = true /\
    is_int32_type type_i = true /\
    int_type_min_max type_lo = Some (int_min_lo, int_max_lo) /\
    int_type_min_max type_i = Some (int_min_i, int_max_i) /\
    int_type_min_max type_hi = Some (int_min_hi, int_max_hi) /\
    int_min = Z.max (Z.max int_min_lo int_min_i) int_min_hi /\
    int_max = Z.min int_max_i int_max_hi /\
    forall m n,
      range m n ->
      m <= int_max_lo /\ int_min <= m <= n /\
      int_min_hi <= n <= int_max.
Proof.
  intros.
  inv H.
  exists int_min_lo, int_max_lo, int_min_i, int_max_i, int_min_hi, int_max_hi, (Z.max (Z.max int_min_lo int_min_i) int_min_hi), (Z.min int_max_i int_max_hi).
  do 7 (split; auto).
  intros.
  destruct (int_max_lo <? Z.min int_max_i int_max_hi) eqn:?H, (int_min_hi >? Z.max (Z.max int_min_lo int_min_i) int_min_hi) eqn:?H.
  + rewrite Z.ltb_lt in H5.
    rewrite Z.gtb_lt in H6.
    omega.
  + rewrite Z.ltb_lt in H5.
    rewrite <- not_true_iff_false, Z.gtb_lt in H6.
    omega.
  + rewrite <- not_true_iff_false, Z.ltb_lt in H5.
    rewrite Z.gtb_lt in H6.
    omega.
  + rewrite <- not_true_iff_false, Z.ltb_lt in H5.
    rewrite <- not_true_iff_false, Z.gtb_lt in H6.
    omega.
Qed.
*)
Inductive range_init_h (type_i type_hi: type): Z -> (Z -> Prop) -> Prop :=
| construct_range_init_h:
  forall int_min int_max,
    int_type_min_max type_i type_hi = Some (int_min, int_max) ->
    range_init_h type_i type_hi int_min (fun n =>
      int_min <= n <= int_max).(*
Lemma range_init_hl_spec: forall (type_lo type_i type_hi: type) range,
  range_init_hl type_lo type_i type_hi range ->
  exists int_min_lo int_max_lo int_min_i int_max_i int_min_hi int_max_hi
    int_min int_max,
    is_int32_type type_lo = true /\
    is_int32_type type_i = true /\
    int_type_min_max type_lo = Some (int_min_lo, int_max_lo) /\
    int_type_min_max type_i = Some (int_min_i, int_max_i) /\
    int_type_min_max type_hi = Some (int_min_hi, int_max_hi) /\
    int_min = Z.max (Z.max int_min_lo int_min_i) int_min_hi /\
    int_max = Z.min int_max_i int_max_hi /\
    forall m n,
      range m n ->
      m <= int_max_lo /\ int_min <= m <= n /\
      int_min_hi <= n <= int_max.
Proof.
  intros.
  inv H.
  exists int_min_lo, int_max_lo, int_min_i, int_max_i, int_min_hi, int_max_hi, (Z.max (Z.max int_min_lo int_min_i) int_min_hi), (Z.min int_max_i int_max_hi).
  do 7 (split; auto).
  intros.
  destruct (int_max_lo <? Z.min int_max_i int_max_hi) eqn:?H, (int_min_hi >? Z.max (Z.max int_min_lo int_min_i) int_min_hi) eqn:?H.
  + rewrite Z.ltb_lt in H5.
    rewrite Z.gtb_lt in H6.
    omega.
  + rewrite Z.ltb_lt in H5.
    rewrite <- not_true_iff_false, Z.gtb_lt in H6.
    omega.
  + rewrite <- not_true_iff_false, Z.ltb_lt in H5.
    rewrite Z.gtb_lt in H6.
    omega.
  + rewrite <- not_true_iff_false, Z.ltb_lt in H5.
    rewrite <- not_true_iff_false, Z.gtb_lt in H6.
    omega.
Qed.
*)
Inductive range_init_h (type_i type_hi: type): Z -> (Z -> Prop) -> Prop :=
| construct_range_init_h:
  forall int_min int_max,
    int_type_min_max type_i type_hi = Some (int_min, int_max) ->
    range_init_h type_i type_hi int_min (fun n =>
      int_min <= n <= int_max).Inductive Int_eqm_unsigned: int -> Z -> Prop :=
| Int_eqm_unsigned_repr: forall z, Int_eqm_unsigned (Int.repr z) z.Inductive Int_eqm_unsigned: int -> Z -> Prop :=
| Int_eqm_unsigned_repr: forall z, Int_eqm_unsigned (Int.repr z) z.Inductive Int64_eqm_unsigned: int64 -> Z -> Prop :=
| Int64_eqm_unsigned_repr: forall z, Int64_eqm_unsigned (Int64.repr z) z.Inductive Int64_eqm_unsigned: int64 -> Z -> Prop :=
| Int64_eqm_unsigned_repr: forall z, Int64_eqm_unsigned (Int64.repr z) z.Inductive Int6432_val: type -> val -> Z -> Prop :=
| Int_64_eqm_unsigned_repr: forall s i64 z,
    Int64_eqm_unsigned i64 z ->
    Int6432_val (Tlong s noattr) (Vlong i64) z
| Int_32_eqm_unsigned_repr: forall s i i32 z,
    Int_eqm_unsigned i32 z ->
    Int6432_val (Tint s i noattr) (Vint i32) z.Inductive Int6432_val: type -> val -> Z -> Prop :=
| Int_64_eqm_unsigned_repr: forall s i64 z,
    Int64_eqm_unsigned i64 z ->
    Int6432_val (Tlong s noattr) (Vlong i64) z
| Int_32_eqm_unsigned_repr: forall s i i32 z,
    Int_eqm_unsigned i32 z ->
    Int6432_val (Tint s i noattr) (Vint i32) z.Inductive Int6432_val: type -> val -> Z -> Prop :=
| Int_64_eqm_unsigned_repr: forall s i64 z,
    Int64_eqm_unsigned i64 z ->
    Int6432_val (Tlong s noattr) (Vlong i64) z
| Int_32_eqm_unsigned_repr: forall s i i32 z,
    Int_eqm_unsigned i32 z ->
    Int6432_val (Tint s i noattr) (Vint i32) z.

(* Specifications: *)
Lemma Int_eqm_unsigned_repr': forall i z, i = Int.repr z -> Int_eqm_unsigned i z.
Lemma Int64_eqm_unsigned_repr': forall i z, i = Int64.repr z -> Int64_eqm_unsigned i z.
Lemma Int_eqm_unsigned_spec: forall i z, Int_eqm_unsigned i z -> Int.eqm (Int.unsigned i) z.
Lemma Int64_eqm_unsigned_spec: forall i z, Int64_eqm_unsigned i z -> Int64.eqm (Int64.unsigned i) z.
Lemma Sfor_inv_rec_step {cs: compspecs} (Delta: tycontext): forall A _i i m hi n assert_callee inv0 inv1, (forall x: A, exists inv0' inv1', Sfor_inv_rec Delta _i i m hi n (assert_callee x) inv0' inv1' /\ inv0 x = inv0' /\ inv1 x = inv1') -> Sfor_inv_rec Delta _i i m hi n (exp assert_callee) (exp inv0) (exp inv1).