Library Coq.Numbers.Cyclic.Int31.Cyclic31

Int31 numbers defines indeed a cyclic structure : Z/(2^31)Z

Author: Arnaud Spiwack (+ Pierre Letouzey)

Require Import List.
Require Import Min.
Require Export Int31.
Require Import Znumtheory.
Require Import Zgcd_alt.
Require Import Zpow_facts.
Require Import CyclicAxioms.
Require Import Lia.

Local Open Scope nat_scope.
Local Open Scope int31_scope.

Local Hint Resolve Z.lt_gt Z.div_pos : zarith.

Section Basics.

Basic results about iszero, shiftl, shiftr

Lemma iszero_eq0 : forall x, iszero x = true -> x =0.
Proof.
destruct x; simpl; intros.
repeat
   match goal with H:(if ?d then _ else _) = true |- _ =>
     destruct d; try discriminate
   end.
reflexivity.
Qed.

Lemma iszero_not_eq0 : forall x, iszero x = false -> x <>0.
Proof.
intros x H Eq; rewrite Eq in H; simpl in *; discriminate.
Qed.

Lemma sneakl_shiftr : forall x,
  x = sneakl (firstr x) (shiftr x).
Proof.
destruct x; simpl; auto.
Qed.

Lemma sneakr_shiftl : forall x,
  x = sneakr (firstl x) (shiftl x).
Proof.
destruct x; simpl; auto.
Qed.

Lemma twice_zero : forall x,
  twice x = 0 <-> twice_plus_one x = 1.
Proof.
destruct x; simpl in *; split;
intro H; injection H; intros; subst; auto.
Qed.

Lemma twice_or_twice_plus_one : forall x,
  x = twice (shiftr x) \/ x = twice_plus_one (shiftr x).
Proof.
intros; case_eq (firstr x); intros.
destruct x; simpl in *; rewrite H; auto.
destruct x; simpl in *; rewrite H; auto.
Qed.

Iterated shift to the right

Definition nshiftr x := nat_rect _ x (fun _ => shiftr).

Lemma nshiftr_S :
  forall n x, nshiftr x (S n) = shiftr (nshiftr x n).
Proof.
reflexivity.
Qed.

Lemma nshiftr_S_tail :
  forall n x, nshiftr x (S n) = nshiftr (shiftr x) n.
Proof.
intros n; elim n; simpl; auto.
intros; now f_equal.
Qed.

Lemma nshiftr_n_0 : forall n, nshiftr 0 n = 0.
Proof.
induction n; simpl; auto.
rewrite IHn; auto.
Qed.

Lemma nshiftr_size : forall x, nshiftr x size = 0.
Proof.
destruct x; simpl; auto.
Qed.

Lemma nshiftr_above_size : forall k x, size <=k ->
  nshiftr x k = 0.
Proof.
intros.
replace k with ((k-size )+size)%nat by omega.
induction (k-size)%nat; auto.
  rewrite nshiftr_size; auto.
  simpl; rewrite IHn; auto.
Qed.

Iterated shift to the left

Definition nshiftl x := nat_rect _ x (fun _ => shiftl).

Lemma nshiftl_S :
  forall n x, nshiftl x (S n) = shiftl (nshiftl x n).
Proof.
reflexivity.
Qed.

Lemma nshiftl_S_tail :
  forall n x, nshiftl x (S n) = nshiftl (shiftl x) n.
Proof.
intros n; elim n; simpl; intros; now f_equal.
Qed.

Lemma nshiftl_n_0 : forall n, nshiftl 0 n = 0.
Proof.
induction n; simpl; auto.
rewrite IHn; auto.
Qed.

Lemma nshiftl_size : forall x, nshiftl x size = 0.
Proof.
destruct x; simpl; auto.
Qed.

Lemma nshiftl_above_size : forall k x, size <=k ->
  nshiftl x k = 0.
Proof.
intros.
replace k with ((k-size )+size)%nat by omega.
induction (k-size)%nat; auto.
  rewrite nshiftl_size; auto.
  simpl; rewrite IHn; auto.
Qed.

Lemma firstr_firstl :
  forall x, firstr x = firstl (nshiftl x (pred size)).
Proof.
destruct x; simpl; auto.
Qed.

Lemma firstl_firstr :
  forall x, firstl x = firstr (nshiftr x (pred size)).
Proof.
destruct x; simpl; auto.
Qed.

More advanced results about nshiftr

Lemma nshiftr_predsize_0_firstl : forall x,
  nshiftr x (pred size) = 0 -> firstl x = D0.
Proof.
destruct x; compute; intros H; injection H; intros; subst; auto.
Qed.

Lemma nshiftr_0_propagates : forall n p x, n <= p ->
  nshiftr x n = 0 -> nshiftr x p = 0.
Proof.
intros.
replace p with ((p-n)+n)%nat by omega.
induction (p-n)%nat.
simpl; auto.
simpl; rewrite IHn0; auto.
Qed.

Lemma nshiftr_0_firstl : forall n x, n < size ->
  nshiftr x n = 0 -> firstl x = D0.
Proof.
intros.
apply nshiftr_predsize_0_firstl.
apply nshiftr_0_propagates with n; auto; omega.
Qed.

Some induction principles over int31

Not used for the moment. Are they really useful ?

Lemma int31_ind_sneakl : forall P : int31 ->Prop,
  P 0 ->
  (forall x d, P x -> P (sneakl d x)) ->
  forall x, P x.
Proof.
intros.
assert (forall n, n <=size -> P (nshiftr x (size - n))).
induction n; intros.
rewrite nshiftr_size; auto.
rewrite sneakl_shiftr.
apply H0.
change (P (nshiftr x (S (size - S n)))).
replace (S (size - S n))%nat with (size - n)%nat by omega.
apply IHn; omega.
change x with (nshiftr x (size -size)); auto.
Qed.

Lemma int31_ind_twice : forall P : int31 ->Prop,
  P 0 ->
  (forall x, P x -> P (twice x)) ->
  (forall x, P x -> P (twice_plus_one x)) ->
  forall x, P x.
Proof.
induction x using int31_ind_sneakl; auto.
destruct d; auto.
Qed.

Some generic results about recr

Section Recr.

recr satisfies the fixpoint equation used for its definition.

Variable (A:Type)(case0:A)(caserec:digits ->int31 ->A ->A).

Lemma recr_aux_eqn : forall n x, iszero x = false ->
   recr_aux (S n) A case0 caserec x =
   caserec (firstr x) (shiftr x) (recr_aux n A case0 caserec (shiftr x)).
Proof.
intros; simpl; rewrite H; auto.
Qed.

Lemma recr_aux_converges :
  forall n p x, n <= size -> n <= p ->
  recr_aux n A case0 caserec (nshiftr x (size - n)) =
  recr_aux p A case0 caserec (nshiftr x (size - n)).
Proof.
induction n.
simpl minus; intros.
rewrite nshiftr_size; destruct p; simpl; auto.
intros.
destruct p.
inversion H0.
unfold recr_aux; fold recr_aux.
destruct (iszero (nshiftr x (size - S n))); auto.
f_equal.
change (shiftr (nshiftr x (size - S n))) with (nshiftr x (S (size - S n))).
replace (S (size - S n))%nat with (size - n)%nat by omega.
apply IHn; auto with arith.
Qed.

Lemma recr_eqn : forall x, iszero x = false ->
  recr A case0 caserec x =
  caserec (firstr x) (shiftr x) (recr A case0 caserec (shiftr x)).
Proof.
intros.
unfold recr.
change x with (nshiftr x (size - size)).
rewrite (recr_aux_converges size (S size)); auto with arith.
rewrite recr_aux_eqn; auto.
Qed.

recr is usually equivalent to a variant recrbis written without iszero check.

Fixpoint recrbis_aux (n:nat)(A:Type)(case0:A)(caserec:digits ->int31 ->A ->A)
(i:int31) : A :=
  match n with
  | O => case0
  | S next =>
      let si := shiftr i in
      caserec (firstr i) si (recrbis_aux next A case0 caserec si)
  end.

Definition recrbis := recrbis_aux size.

Hypothesis case0_caserec : caserec D0 0 case0 = case0.

Lemma recrbis_aux_equiv : forall n x,
   recrbis_aux n A case0 caserec x = recr_aux n A case0 caserec x.
Proof.
induction n; simpl; auto; intros.
case_eq (iszero x); intros; [ | f_equal; auto ].
rewrite (iszero_eq0 _ H); simpl; auto.
replace (recrbis_aux n A case0 caserec 0) with case0; auto.
clear H IHn; induction n; simpl; congruence.
Qed.

Lemma recrbis_equiv : forall x,
   recrbis A case0 caserec x = recr A case0 caserec x.
Proof.
intros; apply recrbis_aux_equiv; auto.
Qed.

End Recr.

Incrementation

Section Incr.

Variant of incr via recrbis

Let Incr (b : digits) (si rec : int31) :=
  match b with
   | D0 => sneakl D1 si
   | D1 => sneakl D0 rec
  end.

Definition incrbis_aux n x := recrbis_aux n _ In Incr x.

Lemma incrbis_aux_equiv : forall x, incrbis_aux size x = incr x.
Proof.
unfold incr, recr, incrbis_aux; fold Incr; intros.
apply recrbis_aux_equiv; auto.
Qed.

Recursive equations satisfied by incr

Lemma incr_eqn1 :
  forall x, firstr x = D0 -> incr x = twice_plus_one (shiftr x).
Proof.
intros.
case_eq (iszero x); intros.
rewrite (iszero_eq0 _ H0); simpl; auto.
unfold incr; rewrite recr_eqn; fold incr; auto.
rewrite H; auto.
Qed.

Lemma incr_eqn2 :
  forall x, firstr x = D1 -> incr x = twice (incr (shiftr x)).
Proof.
intros.
case_eq (iszero x); intros.
rewrite (iszero_eq0 _ H0) in H; simpl in H; discriminate.
unfold incr; rewrite recr_eqn; fold incr; auto.
rewrite H; auto.
Qed.

Lemma incr_twice : forall x, incr (twice x) = twice_plus_one x.
Proof.
intros.
rewrite incr_eqn1; destruct x; simpl; auto.
Qed.

Lemma incr_twice_plus_one_firstl :
  forall x, firstl x = D0 -> incr (twice_plus_one x) = twice (incr x).
Proof.
intros.
rewrite incr_eqn2; [ | destruct x; simpl; auto ].
f_equal; f_equal.
destruct x; simpl in *; rewrite H; auto.
Qed.

The previous result is actually true even without the constraint on firstl, but this is harder to prove (see later).

End Incr.

Conversion to Z : the phi function

Section Phi.

Variant of phi via recrbis

Let Phi := fun b (_:int31) =>
match b with D0 => Z.double | D1 => Z.succ_double end.

Definition phibis_aux n x := recrbis_aux n _ Z0 Phi x.

Lemma phibis_aux_equiv : forall x, phibis_aux size x = phi x.
Proof.
unfold phi, recr, phibis_aux; fold Phi; intros.
apply recrbis_aux_equiv; auto.
Qed.

Recursive equations satisfied by phi

Lemma phi_eqn1 : forall x, firstr x = D0 ->
  phi x = Z.double (phi (shiftr x)).
Proof.
intros.
case_eq (iszero x); intros.
rewrite (iszero_eq0 _ H0); simpl; auto.
intros; unfold phi; rewrite recr_eqn; fold phi; auto.
rewrite H; auto.
Qed.

Lemma phi_eqn2 : forall x, firstr x = D1 ->
  phi x = Z.succ_double (phi (shiftr x)).
Proof.
intros.
case_eq (iszero x); intros.
rewrite (iszero_eq0 _ H0) in H; simpl in H; discriminate.
intros; unfold phi; rewrite recr_eqn; fold phi; auto.
rewrite H; auto.
Qed.

Lemma phi_twice_firstl : forall x, firstl x = D0 ->
  phi (twice x) = Z.double (phi x).
Proof.
intros.
rewrite phi_eqn1; auto; [ | destruct x; auto ].
f_equal; f_equal.
destruct x; simpl in *; rewrite H; auto.
Qed.

Lemma phi_twice_plus_one_firstl : forall x, firstl x = D0 ->
  phi (twice_plus_one x) = Z.succ_double (phi x).
Proof.
intros.
rewrite phi_eqn2; auto; [ | destruct x; auto ].
f_equal; f_equal.
destruct x; simpl in *; rewrite H; auto.
Qed.

End Phi.

phi x is positive and lower than 2^31

Lemma phibis_aux_pos : forall n x, (0 <= phibis_aux n x)%Z.
Proof.
induction n.
simpl; unfold phibis_aux; simpl; auto with zarith.
intros.
unfold phibis_aux, recrbis_aux; fold recrbis_aux;
  fold (phibis_aux n (shiftr x)).
destruct (firstr x).
specialize IHn with (shiftr x); rewrite Z.double_spec; omega.
specialize IHn with (shiftr x); rewrite Z.succ_double_spec; omega.
Qed.

Lemma phibis_aux_bounded :
  forall n x, n <= size ->
  (phibis_aux n (nshiftr x (size -n)) < 2 ^ (Z.of_nat n ))%Z.
Proof.
induction n.
simpl minus; unfold phibis_aux; simpl; auto with zarith.
intros.
unfold phibis_aux, recrbis_aux; fold recrbis_aux;
  fold (phibis_aux n (shiftr (nshiftr x (size - S n)))).
assert (shiftr (nshiftr x (size - S n)) = nshiftr x (size -n)).
  replace (size - n)%nat with (S (size - (S n))) by omega.
  simpl; auto.
rewrite H0.
assert (H1 : n <= size) by omega.
specialize (IHn x H1).
set (y:=phibis_aux n (nshiftr x (size - n))) in *.
rewrite Nat2Z.inj_succ, Z.pow_succ_r; auto with zarith.
case_eq (firstr (nshiftr x (size - S n))); intros.
rewrite Z.double_spec; auto with zarith.
rewrite Z.succ_double_spec; auto with zarith.
Qed.

Lemma phi_nonneg : forall x, (0 <= phi x)%Z.
Proof.
intros.
rewrite <- phibis_aux_equiv.
apply phibis_aux_pos.
Qed.

Hint Resolve phi_nonneg : zarith.

Lemma phi_bounded : forall x, (0 <= phi x < 2 ^ (Z.of_nat size ))%Z.
Proof.
intros. split; [auto with zarith|].
rewrite <- phibis_aux_equiv.
change x with (nshiftr x (size -size)).
apply phibis_aux_bounded; auto.
Qed.

Lemma phibis_aux_lowerbound :
  forall n x, firstr (nshiftr x n) = D1 ->
  (2 ^ Z.of_nat n <= phibis_aux (S n) x)%Z.
Proof.
induction n.
intros.
unfold nshiftr in H; simpl in *.
unfold phibis_aux, recrbis_aux.
rewrite H, Z.succ_double_spec; omega.

intros.
remember (S n) as m.
unfold phibis_aux, recrbis_aux; fold recrbis_aux;
  fold (phibis_aux m (shiftr x)).
subst m.
rewrite Nat2Z.inj_succ, Z.pow_succ_r; auto with zarith.
assert (2^(Z.of_nat n) <= phibis_aux (S n) (shiftr x))%Z.
  apply IHn.
  rewrite <- nshiftr_S_tail; auto.
destruct (firstr x).
change (Z.double (phibis_aux (S n) (shiftr x))) with
        (2*(phibis_aux (S n) (shiftr x)))%Z.
omega.
rewrite Z.succ_double_spec; omega.
Qed.

Lemma phi_lowerbound :
  forall x, firstl x = D1 -> (2^(Z.of_nat (pred size)) <= phi x)%Z.
Proof.
intros.
generalize (phibis_aux_lowerbound (pred size) x).
rewrite <- firstl_firstr.
change (S (pred size)) with size; auto.
rewrite phibis_aux_equiv; auto.
Qed.

Equivalence modulo 2^n

Section EqShiftL.

After killing n bits at the left, are the numbers equal ?

Definition EqShiftL n x y :=
  nshiftl x n = nshiftl y n.

Lemma EqShiftL_zero : forall x y, EqShiftL O x y <-> x = y.
Proof.
unfold EqShiftL; intros; unfold nshiftl; simpl; split; auto.
Qed.

Lemma EqShiftL_size : forall k x y, size <=k -> EqShiftL k x y.
Proof.
red; intros; rewrite 2 nshiftl_above_size; auto.
Qed.

Lemma EqShiftL_le : forall k k' x y, k <= k' ->
   EqShiftL k x y -> EqShiftL k' x y.
Proof.
unfold EqShiftL; intros.
replace k' with ((k'-k)+k)%nat by omega.
remember (k'-k)%nat as n.
clear Heqn H k'.
induction n; simpl; auto.
f_equal; auto.
Qed.

Lemma EqShiftL_firstr : forall k x y, k < size ->
  EqShiftL k x y -> firstr x = firstr y.
Proof.
intros.
rewrite 2 firstr_firstl.
f_equal.
apply EqShiftL_le with k; auto.
unfold size.
auto with arith.
Qed.

Lemma EqShiftL_twice : forall k x y,
  EqShiftL k (twice x) (twice y) <-> EqShiftL (S k) x y.
Proof.
intros; unfold EqShiftL.
rewrite 2 nshiftl_S_tail; split; auto.
Qed.

From int31 to list of digits.

Lower (=rightmost) bits comes first.

Definition i2l := recrbis _ nil (fun d _ rec => d ::rec).

Lemma i2l_length : forall x, length (i2l x) = size.
Proof.
intros; reflexivity.
Qed.

Fixpoint lshiftl l x :=
   match l with
     | nil => x
     | d::l => sneakl d (lshiftl l x)
   end.

Definition l2i l := lshiftl l On.

Lemma l2i_i2l : forall x, l2i (i2l x) = x.
Proof.
destruct x; compute; auto.
Qed.

Lemma i2l_sneakr : forall x d,
   i2l (sneakr d x) = tail (i2l x) ++ d ::nil.
Proof.
destruct x; compute; auto.
Qed.

Lemma i2l_sneakl : forall x d,
   i2l (sneakl d x) = d :: removelast (i2l x).
Proof.
destruct x; compute; auto.
Qed.

Lemma i2l_l2i : forall l, length l = size ->
  i2l (l2i l) = l.
Proof.
repeat (destruct l as [ |? l]; [intros; discriminate | ]).
destruct l; [ | intros; discriminate].
intros _; compute; auto.
Qed.

Fixpoint cstlist (A:Type)(a:A) n :=
  match n with
   | O => nil
   | S n => a ::cstlist _ a n
  end.

Lemma i2l_nshiftl : forall n x, n <=size ->
  i2l (nshiftl x n) = cstlist _ D0 n ++ firstn (size -n) (i2l x).
Proof.
induction n.
intros.
assert (firstn (size -0) (i2l x) = i2l x).
  rewrite <- minus_n_O, <- (i2l_length x).
  induction (i2l x); simpl; f_equal; auto.
rewrite H0; clear H0.
reflexivity.

intros.
rewrite nshiftl_S.
unfold shiftl; rewrite i2l_sneakl.
simpl cstlist.
rewrite <- app_comm_cons; f_equal.
rewrite IHn; [ | omega].
rewrite removelast_app.
apply f_equal.
replace (size -n)%nat with (S (size - S n))%nat by omega.
rewrite removelast_firstn; auto.
rewrite i2l_length; omega.
generalize (firstn_length (size -n) (i2l x)).
rewrite i2l_length.
intros H0 H1. rewrite H1 in H0.
rewrite min_l in H0 by omega.
simpl length in H0.
omega.
Qed.

i2l can be used to define a relation equivalent to EqShiftL

Lemma EqShiftL_i2l : forall k x y,
   EqShiftL k x y <-> firstn (size -k) (i2l x) = firstn (size -k) (i2l y).
Proof.
intros.
destruct (le_lt_dec size k) as [Hle|Hlt].
split; intros.
replace (size -k)%nat with O by omega.
unfold firstn; auto.
apply EqShiftL_size; auto.

unfold EqShiftL.
assert (k <= size) by omega.
split; intros.
assert (i2l (nshiftl x k) = i2l (nshiftl y k)) by (f_equal; auto).
rewrite 2 i2l_nshiftl in H1; auto.
eapply app_inv_head; eauto.
assert (i2l (nshiftl x k) = i2l (nshiftl y k)).
  rewrite 2 i2l_nshiftl; auto.
  f_equal; auto.
rewrite <- (l2i_i2l (nshiftl x k)), <- (l2i_i2l (nshiftl y k)).
f_equal; auto.
Qed.

This equivalence allows proving easily the following delicate result

Lemma EqShiftL_twice_plus_one : forall k x y,
  EqShiftL k (twice_plus_one x) (twice_plus_one y) <-> EqShiftL (S k) x y.
Proof.
intros.
destruct (le_lt_dec size k) as [Hle|Hlt].
split; intros; apply EqShiftL_size; auto.

rewrite 2 EqShiftL_i2l.
unfold twice_plus_one.
rewrite 2 i2l_sneakl.
replace (size -k)%nat with (S (size - S k))%nat by omega.
remember (size - S k)%nat as n.
remember (i2l x) as lx.
remember (i2l y) as ly.
simpl.
rewrite 2 firstn_removelast.
split; intros.
injection H; auto.
f_equal; auto.
subst ly n; rewrite i2l_length; omega.
subst lx n; rewrite i2l_length; omega.
Qed.

Lemma EqShiftL_shiftr : forall k x y, EqShiftL k x y ->
  EqShiftL (S k) (shiftr x) (shiftr y).
Proof.
intros.
destruct (le_lt_dec size (S k)) as [Hle|Hlt].
apply EqShiftL_size; auto.
case_eq (firstr x); intros.
rewrite <- EqShiftL_twice.
unfold twice; rewrite <- H0.
rewrite <- sneakl_shiftr.
rewrite (EqShiftL_firstr k x y); auto.
rewrite <- sneakl_shiftr; auto.
omega.
rewrite <- EqShiftL_twice_plus_one.
unfold twice_plus_one; rewrite <- H0.
rewrite <- sneakl_shiftr.
rewrite (EqShiftL_firstr k x y); auto.
rewrite <- sneakl_shiftr; auto.
omega.
Qed.

Lemma EqShiftL_incrbis : forall n k x y, n <=size ->
  (n +k =S size)%nat ->
  EqShiftL k x y ->
  EqShiftL k (incrbis_aux n x) (incrbis_aux n y).
Proof.
induction n; simpl; intros.
red; auto.
destruct (eq_nat_dec k size).
  subst k; apply EqShiftL_size; auto.
unfold incrbis_aux; simpl;
  fold (incrbis_aux n (shiftr x)); fold (incrbis_aux n (shiftr y)).

rewrite (EqShiftL_firstr k x y); auto; try omega.
case_eq (firstr y); intros.
rewrite EqShiftL_twice_plus_one.
apply EqShiftL_shiftr; auto.

rewrite EqShiftL_twice.
apply IHn; try omega.
apply EqShiftL_shiftr; auto.
Qed.

Lemma EqShiftL_incr : forall x y,
  EqShiftL 1 x y -> EqShiftL 1 (incr x) (incr y).
Proof.
intros.
rewrite <- 2 incrbis_aux_equiv.
apply EqShiftL_incrbis; auto.
Qed.

End EqShiftL.

More equations about incr

Lemma incr_twice_plus_one :
forall x, incr (twice_plus_one x) = twice (incr x).
Proof.
intros.
rewrite incr_eqn2; [ | destruct x; simpl; auto].
apply EqShiftL_incr.
red; destruct x; simpl; auto.
Qed.

Lemma incr_firstr : forall x, firstr (incr x) <> firstr x.
Proof.
intros.
case_eq (firstr x); intros.
rewrite incr_eqn1; auto.
destruct (shiftr x); simpl; discriminate.
rewrite incr_eqn2; auto.
destruct (incr (shiftr x)); simpl; discriminate.
Qed.

Lemma incr_inv : forall x y,
incr x = twice_plus_one y -> x = twice y.
Proof.
intros.
case_eq (iszero x); intros.
rewrite (iszero_eq0 _ H0) in *; simpl in *.
change (incr 0) with 1 in H.
symmetry; rewrite twice_zero; auto.
case_eq (firstr x); intros.
rewrite incr_eqn1 in H; auto.
clear H0; destruct x; destruct y; simpl in *.
injection H; intros; subst; auto.
elim (incr_firstr x).
rewrite H1, H; destruct y; simpl; auto.
Qed.

Conversion from Z : the phi_inv function

First, recursive equations

Lemma phi_inv_double_plus_one : forall z,
   phi_inv (Z.succ_double z) = twice_plus_one (phi_inv z).
Proof.
destruct z; simpl; auto.
induction p; simpl.
rewrite 2 incr_twice; auto.
rewrite incr_twice, incr_twice_plus_one.
f_equal.
apply incr_inv; auto.
auto.
Qed.

Lemma phi_inv_double : forall z,
   phi_inv (Z.double z) = twice (phi_inv z).
Proof.
destruct z; simpl; auto.
rewrite incr_twice_plus_one; auto.
Qed.

Lemma phi_inv_incr : forall z,
  phi_inv (Z.succ z) = incr (phi_inv z).
Proof.
destruct z.
simpl; auto.
simpl; auto.
induction p; simpl; auto.
rewrite <- Pos.add_1_r, IHp, incr_twice_plus_one; auto.
rewrite incr_twice; auto.
simpl; auto.
destruct p; simpl; auto.
rewrite incr_twice; auto.
f_equal.
rewrite incr_twice_plus_one; auto.
induction p; simpl; auto.
rewrite incr_twice; auto.
f_equal.
rewrite incr_twice_plus_one; auto.
Qed.

phi_inv o inv, the always-exact and easy-to-prove trip : from int31 to Z and then back to int31.

Lemma phi_inv_phi_aux :
  forall n x, n <= size ->
   phi_inv (phibis_aux n (nshiftr x (size -n))) =
   nshiftr x (size -n).
Proof.
induction n.
intros; simpl minus.
rewrite nshiftr_size; auto.
intros.
unfold phibis_aux, recrbis_aux; fold recrbis_aux;
  fold (phibis_aux n (shiftr (nshiftr x (size -S n)))).
assert (shiftr (nshiftr x (size - S n)) = nshiftr x (size -n)).
  replace (size - n)%nat with (S (size - (S n))); auto; omega.
rewrite H0.
case_eq (firstr (nshiftr x (size - S n))); intros.

rewrite phi_inv_double.
rewrite IHn by omega.
rewrite <- H0.
remember (nshiftr x (size - S n)) as y.
destruct y; simpl in H1; rewrite H1; auto.

rewrite phi_inv_double_plus_one.
rewrite IHn by omega.
rewrite <- H0.
remember (nshiftr x (size - S n)) as y.
destruct y; simpl in H1; rewrite H1; auto.
Qed.

Lemma phi_inv_phi : forall x, phi_inv (phi x) = x.
Proof.
intros.
rewrite <- phibis_aux_equiv.
replace x with (nshiftr x (size - size)) by auto.
apply phi_inv_phi_aux; auto.
Qed.

The other composition phi o phi_inv is harder to prove correct. In particular, an overflow can happen, so a modulo is needed. For the moment, we proceed via several steps, the first one being a detour to positive_to_in31.

positive_to_int31

A variant of p2i with twice and twice_plus_one instead of 2*i and 2*i+1

Fixpoint p2ibis n p : (N *int31)%type :=
  match n with
    | O => (Npos p , On )
    | S n => match p with
               | xO p => let (r,i) := p2ibis n p in (r , twice i )
               | xI p => let (r,i) := p2ibis n p in (r , twice_plus_one i )
               | xH => (N0 , In )
             end
  end.

Lemma p2ibis_bounded : forall n p,
  nshiftr (snd (p2ibis n p)) n = 0.
Proof.
induction n.
simpl; intros; auto.
simpl p2ibis; intros.
destruct p; simpl snd.

specialize IHn with p.
destruct (p2ibis n p). simpl @snd in *.
rewrite nshiftr_S_tail.
destruct (le_lt_dec size n) as [Hle|Hlt].
rewrite nshiftr_above_size; auto.
assert (H:=nshiftr_0_firstl _ _ Hlt IHn).
replace (shiftr (twice_plus_one i)) with i; auto.
destruct i; simpl in *. rewrite H; auto.

specialize IHn with p.
destruct (p2ibis n p); simpl @snd in *.
rewrite nshiftr_S_tail.
destruct (le_lt_dec size n) as [Hle|Hlt].
rewrite nshiftr_above_size; auto.
assert (H:=nshiftr_0_firstl _ _ Hlt IHn).
replace (shiftr (twice i)) with i; auto.
destruct i; simpl in *; rewrite H; auto.

rewrite nshiftr_S_tail; auto.
replace (shiftr In) with 0; auto.
apply nshiftr_n_0.
Qed.

Local Open Scope Z_scope.

Lemma p2ibis_spec : forall n p, (n <=size)%nat ->
    Zpos p = (Z.of_N (fst (p2ibis n p)))*2^(Z.of_nat n ) +
             phi (snd (p2ibis n p)).
Proof.
induction n; intros.
simpl; rewrite Pos.mul_1_r; auto.
replace (2^(Z.of_nat (S n)))%Z with (2*2^(Z.of_nat n))%Z by
  (rewrite <- Z.pow_succ_r, <- Zpos_P_of_succ_nat;
   auto with zarith).
rewrite (Z.mul_comm 2).
assert (n<=size)%nat by omega.
destruct p; simpl; [ | | auto];
  specialize (IHn p H0);
  generalize (p2ibis_bounded n p);
  destruct (p2ibis n p) as (r,i); simpl in *; intros.

change (Zpos p~1) with (2*Zpos p + 1)%Z.
rewrite phi_twice_plus_one_firstl, Z.succ_double_spec.
rewrite IHn; ring.
apply (nshiftr_0_firstl n); auto; try omega.

change (Zpos p~0) with (2*Zpos p)%Z.
rewrite phi_twice_firstl.
change (Z.double (phi i)) with (2*(phi i))%Z.
rewrite IHn; ring.
apply (nshiftr_0_firstl n); auto; try omega.
Qed.

We now prove that this p2ibis is related to phi_inv_positive

Lemma phi_inv_positive_p2ibis : forall n p, (n <=size)%nat ->
  EqShiftL (size -n) (phi_inv_positive p) (snd (p2ibis n p)).
Proof.
induction n.
intros.
apply EqShiftL_size; auto.
intros.
simpl p2ibis; destruct p; [ | | red; auto];
  specialize IHn with p;
  destruct (p2ibis n p); simpl @snd in *; simpl phi_inv_positive;
  rewrite ?EqShiftL_twice_plus_one, ?EqShiftL_twice;
  replace (S (size - S n))%nat with (size - n)%nat by omega;
  apply IHn; omega.
Qed.

This gives the expected result about phi o phi_inv, at least for the positive case.

Lemma phi_phi_inv_positive : forall p,
phi (phi_inv_positive p) = (Zpos p ) mod (2^(Z.of_nat size )).
Proof.
intros.
replace (phi_inv_positive p) with (snd (p2ibis size p)).
rewrite (p2ibis_spec size p) by auto.
rewrite Z.add_comm, Z_mod_plus.
symmetry; apply Zmod_small.
apply phi_bounded.
auto with zarith.
symmetry.
rewrite <- EqShiftL_zero.
apply (phi_inv_positive_p2ibis size p); auto.
Qed.

Moreover, p2ibis is also related with p2i and hence with positive_to_int31.

Lemma double_twice_firstl : forall x, firstl x = D0 ->
  (Twon *x = twice x)%int31.
Proof.
intros.
unfold mul31.
rewrite <- Z.double_spec, <- phi_twice_firstl, phi_inv_phi; auto.
Qed.

Lemma double_twice_plus_one_firstl : forall x, firstl x = D0 ->
  (Twon *x +In = twice_plus_one x)%int31.
Proof.
intros.
rewrite double_twice_firstl; auto.
unfold add31.
rewrite phi_twice_firstl, <- Z.succ_double_spec,
   <- phi_twice_plus_one_firstl, phi_inv_phi; auto.
Qed.

Lemma p2i_p2ibis : forall n p, (n <=size)%nat ->
  p2i n p = p2ibis n p.
Proof.
induction n; simpl; auto; intros.
destruct p; auto; specialize IHn with p;
  generalize (p2ibis_bounded n p);
  rewrite IHn; try omega; destruct (p2ibis n p); simpl; intros;
  f_equal; auto.
apply double_twice_plus_one_firstl.
apply (nshiftr_0_firstl n); auto; omega.
apply double_twice_firstl.
apply (nshiftr_0_firstl n); auto; omega.
Qed.

Lemma positive_to_int31_phi_inv_positive : forall p,
   snd (positive_to_int31 p) = phi_inv_positive p.
Proof.
intros; unfold positive_to_int31.
rewrite p2i_p2ibis; auto.
symmetry.
rewrite <- EqShiftL_zero.
apply (phi_inv_positive_p2ibis size); auto.
Qed.

Lemma positive_to_int31_spec : forall p,
    Zpos p = (Z.of_N (fst (positive_to_int31 p)))*2^(Z.of_nat size ) +
               phi (snd (positive_to_int31 p)).
Proof.
unfold positive_to_int31.
intros; rewrite p2i_p2ibis; auto.
apply p2ibis_spec; auto.
Qed.

Thanks to the result about phi o phi_inv_positive, we can now establish easily the most general results about phi o twice and so one.

Lemma phi_twice : forall x,
   phi (twice x) = (Z.double (phi x)) mod 2^(Z.of_nat size ).
Proof.
intros.
pattern x at 1; rewrite <- (phi_inv_phi x).
rewrite <- phi_inv_double.
assert (0 <= Z.double (phi x)).
  rewrite Z.double_spec; generalize (phi_bounded x); omega.
destruct (Z.double (phi x)).
simpl; auto.
apply phi_phi_inv_positive.
compute in H; elim H; auto.
Qed.

Lemma phi_twice_plus_one : forall x,
   phi (twice_plus_one x) = (Z.succ_double (phi x)) mod 2^(Z.of_nat size ).
Proof.
intros.
pattern x at 1; rewrite <- (phi_inv_phi x).
rewrite <- phi_inv_double_plus_one.
assert (0 <= Z.succ_double (phi x)).
  rewrite Z.succ_double_spec; generalize (phi_bounded x); omega.
destruct (Z.succ_double (phi x)).
simpl; auto.
apply phi_phi_inv_positive.
compute in H; elim H; auto.
Qed.

Lemma phi_incr : forall x,
   phi (incr x) = (Z.succ (phi x)) mod 2^(Z.of_nat size ).
Proof.
intros.
pattern x at 1; rewrite <- (phi_inv_phi x).
rewrite <- phi_inv_incr.
assert (0 <= Z.succ (phi x)).
  change (Z.succ (phi x)) with ((phi x)+1)%Z;
   generalize (phi_bounded x); omega.
destruct (Z.succ (phi x)).
simpl; auto.
apply phi_phi_inv_positive.
compute in H; elim H; auto.
Qed.

With the previous results, we can deal with phi o phi_inv even in the negative case

Lemma phi_phi_inv_negative :
  forall p, phi (incr (complement_negative p)) = (Zneg p ) mod 2^(Z.of_nat size ).
Proof.
induction p.

simpl complement_negative.
rewrite phi_incr in IHp.
rewrite incr_twice, phi_twice_plus_one.
remember (phi (complement_negative p)) as q.
rewrite Z.succ_double_spec.
replace (2*q+1) with (2*(Z.succ q)-1) by omega.
rewrite <- Zminus_mod_idemp_l, <- Zmult_mod_idemp_r, IHp.
rewrite Zmult_mod_idemp_r, Zminus_mod_idemp_l; auto with zarith.

simpl complement_negative.
rewrite incr_twice_plus_one, phi_twice.
remember (phi (incr (complement_negative p))) as q.
rewrite Z.double_spec, IHp, Zmult_mod_idemp_r; auto with zarith.

simpl; auto.
Qed.

Lemma phi_phi_inv :
  forall z, phi (phi_inv z) = z mod 2 ^ (Z.of_nat size ).
Proof.
destruct z.
simpl; auto.
apply phi_phi_inv_positive.
apply phi_phi_inv_negative.
Qed.

End Basics.

Instance int31_ops : ZnZ.Ops int31 :=
{
digits := 31%positive;
zdigits := 31;
to_Z := phi;
of_pos := positive_to_int31;
head0 := head031;
tail0 := tail031;
zero := 0;
one := 1;
minus_one := Tn;
compare := compare31;
eq0 := fun i => match i ?= 0 with Eq => true | _ => false end;
opp_c := fun i => 0 -c i;
opp := opp31;
opp_carry := fun i => 0-i -1;
succ_c := fun i => i +c 1;
add_c := add31c;
add_carry_c := add31carryc;
succ := fun i => i + 1;
add := add31;
add_carry := fun i j => i + j + 1;
pred_c := fun i => i -c 1;
sub_c := sub31c;
sub_carry_c := sub31carryc;
pred := fun i => i - 1;
sub := sub31;
sub_carry := fun i j => i - j - 1;
mul_c := mul31c;
mul := mul31;
square_c := fun x => x *c x;
div21 := div3121;
div_gt := div31;
div := div31;
modulo_gt := fun i j => let (_,r) := i /j in r;
modulo := fun i j => let (_,r) := i /j in r;
gcd_gt := gcd31;
gcd := gcd31;
add_mul_div := addmuldiv31;
pos_mod :=
  fun p i =>
  match p ?= 31 with
    | Lt => addmuldiv31 p 0 (addmuldiv31 (31-p) i 0)
    | _ => i
  end;
is_even :=
  fun i => let (_,r) := i /2 in
  match r ?= 0 with Eq => true | _ => false end;
sqrt2 := sqrt312;
sqrt := sqrt31;
lor := lor31;
land := land31;
lxor := lxor31
}.

Section Int31_Specs.

Local Open Scope Z_scope.

Notation "[| x |]" := (phi x) (at level 0, x at level 99).

Local Notation wB := (2 ^ (Z.of_nat size )).

Lemma wB_pos : wB > 0.
Proof.
  auto with zarith.
Qed.

Notation "[+| c |]" :=
   (interp_carry 1 wB phi c) (at level 0, c at level 99).

Notation "[-| c |]" :=
   (interp_carry (-1) wB phi c) (at level 0, c at level 99).

Notation "[|| x ||]" :=
   (zn2z_to_Z wB phi x) (at level 0, x at level 99).

Lemma spec_zdigits : [| 31 |] = 31.
Proof.
  reflexivity.
Qed.

Lemma spec_more_than_1_digit: 1 < 31.
Proof.
  auto with zarith.
Qed.

Lemma spec_0 : [| 0 |] = 0.
Proof.
  reflexivity.
Qed.

Lemma spec_1 : [| 1 |] = 1.
Proof.
  reflexivity.
Qed.

Lemma spec_m1 : [| Tn |] = wB - 1.
Proof.
  reflexivity.
Qed.

Lemma spec_compare : forall x y,
   (x ?= y)%int31 = ([|x |] ?= [|y |]).
Proof. reflexivity. Qed.

Addition

Lemma spec_add_c : forall x y, [+|add31c x y |] = [|x |] + [|y |].
Proof.
intros; unfold add31c, add31, interp_carry; rewrite phi_phi_inv.
generalize (phi_bounded x)(phi_bounded y); intros.
set (X:=[|x|]) in *; set (Y:=[|y|]) in *; clearbody X Y.

assert ((X+Y) mod wB ?= X+Y <> Eq -> [+|C1 (phi_inv (X+Y))|] = X+Y).
  unfold interp_carry; rewrite phi_phi_inv, Z.compare_eq_iff; intros.
  destruct (Z_lt_le_dec (X+Y) wB).
  contradict H1; auto using Zmod_small with zarith.
  rewrite <- (Z_mod_plus_full (X+Y) (-1) wB).
  rewrite Zmod_small; lia.

generalize (Z.compare_eq ((X+Y) mod wB) (X+Y)); intros Heq.
destruct Z.compare; intros;
  [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1].
Qed.

Lemma spec_succ_c : forall x, [+|add31c x 1|] = [|x |] + 1.
Proof.
intros; apply spec_add_c.
Qed.

Lemma spec_add_carry_c : forall x y, [+|add31carryc x y |] = [|x |] + [|y |] + 1.
Proof.
intros.
unfold add31carryc, interp_carry; rewrite phi_phi_inv.
generalize (phi_bounded x)(phi_bounded y); intros.
set (X:=[|x|]) in *; set (Y:=[|y|]) in *; clearbody X Y.

assert ((X+Y+1) mod wB ?= X+Y+1 <> Eq -> [+|C1 (phi_inv (X+Y+1))|] = X+Y+1).
  unfold interp_carry; rewrite phi_phi_inv, Z.compare_eq_iff; intros.
  destruct (Z_lt_le_dec (X+Y+1) wB).
  contradict H1; auto using Zmod_small with zarith.
  rewrite <- (Z_mod_plus_full (X+Y+1) (-1) wB).
  rewrite Zmod_small; lia.

generalize (Z.compare_eq ((X+Y+1) mod wB) (X+Y+1)); intros Heq.
destruct Z.compare; intros;
  [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1].
Qed.

Lemma spec_add : forall x y, [|x +y |] = ([|x |] + [|y |]) mod wB.
Proof.
intros; apply phi_phi_inv.
Qed.

Lemma spec_add_carry :
         forall x y, [|x +y +1|] = ([|x |] + [|y |] + 1) mod wB.
Proof.
unfold add31; intros.
repeat rewrite phi_phi_inv.
apply Zplus_mod_idemp_l.
Qed.

Lemma spec_succ : forall x, [|x +1|] = ([|x |] + 1) mod wB.
Proof.
intros; rewrite <- spec_1; apply spec_add.
Qed.

Substraction

Lemma spec_sub_c : forall x y, [-|sub31c x y |] = [|x |] - [|y |].
Proof.
unfold sub31c, sub31, interp_carry; intros.
rewrite phi_phi_inv.
generalize (phi_bounded x)(phi_bounded y); intros.
set (X:=[|x|]) in *; set (Y:=[|y|]) in *; clearbody X Y.

assert ((X-Y) mod wB ?= X-Y <> Eq -> [-|C1 (phi_inv (X-Y))|] = X-Y).
  unfold interp_carry; rewrite phi_phi_inv, Z.compare_eq_iff; intros.
  destruct (Z_lt_le_dec (X-Y) 0).
  rewrite <- (Z_mod_plus_full (X-Y) 1 wB).
  rewrite Zmod_small; lia.
  contradict H1; apply Zmod_small; lia.

generalize (Z.compare_eq ((X-Y) mod wB) (X-Y)); intros Heq.
destruct Z.compare; intros;
  [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1].
Qed.

Lemma spec_sub_carry_c : forall x y, [-|sub31carryc x y |] = [|x |] - [|y |] - 1.
Proof.
unfold sub31carryc, sub31, interp_carry; intros.
rewrite phi_phi_inv.
generalize (phi_bounded x)(phi_bounded y); intros.
set (X:=[|x|]) in *; set (Y:=[|y|]) in *; clearbody X Y.

assert ((X-Y-1) mod wB ?= X-Y-1 <> Eq -> [-|C1 (phi_inv (X-Y-1))|] = X-Y-1).
  unfold interp_carry; rewrite phi_phi_inv, Z.compare_eq_iff; intros.
  destruct (Z_lt_le_dec (X-Y-1) 0).
  rewrite <- (Z_mod_plus_full (X-Y-1) 1 wB).
  rewrite Zmod_small; lia.
  contradict H1; apply Zmod_small; lia.

generalize (Z.compare_eq ((X-Y-1) mod wB) (X-Y-1)); intros Heq.
destruct Z.compare; intros;
  [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1].
Qed.

Lemma spec_sub : forall x y, [|x -y |] = ([|x |] - [|y |]) mod wB.
Proof.
intros; apply phi_phi_inv.
Qed.

Lemma spec_sub_carry :
   forall x y, [|x -y -1|] = ([|x |] - [|y |] - 1) mod wB.
Proof.
unfold sub31; intros.
repeat rewrite phi_phi_inv.
apply Zminus_mod_idemp_l.
Qed.

Lemma spec_opp_c : forall x, [-|sub31c 0 x |] = -[|x |].
Proof.
intros; apply spec_sub_c.
Qed.

Lemma spec_opp : forall x, [|0 - x |] = (-[|x |]) mod wB.
Proof.
intros; apply phi_phi_inv.
Qed.

Lemma spec_opp_carry : forall x, [|0 - x - 1|] = wB - [|x |] - 1.
Proof.
unfold sub31; intros.
repeat rewrite phi_phi_inv.
change [|1|] with 1; change [|0|] with 0.
rewrite <- (Z_mod_plus_full (0-[|x|]) 1 wB).
rewrite Zminus_mod_idemp_l.
rewrite Zmod_small; generalize (phi_bounded x); lia.
Qed.

Lemma spec_pred_c : forall x, [-|sub31c x 1|] = [|x |] - 1.
Proof.
intros; apply spec_sub_c.
Qed.

Lemma spec_pred : forall x, [|x -1|] = ([|x |] - 1) mod wB.
Proof.
intros; apply spec_sub.
Qed.

Multiplication

Lemma phi2_phi_inv2 : forall x, [||phi_inv2 x ||] = x mod (wB ^2).
Proof.
assert (forall z, (z / wB ) mod wB * wB + z mod wB = z mod wB ^ 2).
  intros.
  assert ((z/wB ) mod wB = z/wB - (z/wB /wB )*wB).
   rewrite (Z_div_mod_eq (z/wB) wB wB_pos) at 2; ring.
  assert (z mod wB = z - (z/wB )*wB).
   rewrite (Z_div_mod_eq z wB wB_pos) at 2; ring.
  rewrite H.
  rewrite H0 at 1.
  ring_simplify.
  rewrite Zdiv_Zdiv; auto with zarith.
  rewrite (Z_div_mod_eq z (wB *wB)) at 2; auto with zarith.
  change (wB *wB) with (wB ^2); ring.

unfold phi_inv2.
destruct x; unfold zn2z_to_Z; rewrite ?phi_phi_inv;
  change base with wB; auto.
Qed.

Lemma spec_mul_c : forall x y, [|| mul31c x y ||] = [|x |] * [|y |].
Proof.
unfold mul31c; intros.
rewrite phi2_phi_inv2.
apply Zmod_small.
generalize (phi_bounded x)(phi_bounded y); intros.
change (wB ^2) with (wB * wB).
auto using Z.mul_lt_mono_nonneg with zarith.
Qed.

Lemma spec_mul : forall x y, [|x *y |] = ([|x |] * [|y |]) mod wB.
Proof.
intros; apply phi_phi_inv.
Qed.

Lemma spec_square_c : forall x, [|| mul31c x x ||] = [|x |] * [|x |].
Proof.
intros; apply spec_mul_c.
Qed.

Division

Lemma spec_div21 : forall a1 a2 b,
      wB /2 <= [|b |] ->
      [|a1 |] < [|b |] ->
      let (q,r) := div3121 a1 a2 b in
      [|a1 |] *wB + [|a2 |] = [|q |] * [|b |] + [|r |] /\
      0 <= [|r |] < [|b |].
Proof.
unfold div3121; intros.
generalize (phi_bounded a1)(phi_bounded a2)(phi_bounded b); intros.
assert ([|b|]>0) by (auto with zarith).
generalize (Z_div_mod (phi2 a1 a2) [|b|] H4) (Z_div_pos (phi2 a1 a2) [|b|] H4).
unfold Z.div; destruct (Z.div_eucl (phi2 a1 a2) [|b|]).
rewrite ?phi_phi_inv.
destruct 1; intros.
unfold phi2 in *.
change base with wB; change base with wB in H5.
change (Z.pow_pos 2 31) with wB; change (Z.pow_pos 2 31) with wB in H.
rewrite H5, Z.mul_comm.
replace (z0 mod wB) with z0 by (symmetry; apply Zmod_small; omega).
replace (z mod wB) with z; auto with zarith.
  symmetry; apply Zmod_small.
  split.
  apply H7; change base with wB; auto with zarith.
  apply Z.mul_lt_mono_pos_r with [|b|]; [omega| ].
  rewrite Z.mul_comm.
  apply Z.le_lt_trans with ([|b|]*z+z0); [omega| ].
  rewrite <- H5.
  apply Z.le_lt_trans with ([|a1|]*wB +(wB -1)); [omega | ].
  replace ([|a1|]*wB +(wB -1)) with (wB *([|a1|]+1)-1) by ring.
  assert (wB *([|a1|]+1) <= wB *[|b|]); try omega.
  apply Z.mul_le_mono_nonneg; omega.
Qed.

Lemma spec_div : forall a b, 0 < [|b |] ->
      let (q,r) := div31 a b in
      [|a |] = [|q |] * [|b |] + [|r |] /\
      0 <= [|r |] < [|b |].
Proof.
unfold div31; intros.
assert ([|b|]>0) by (auto with zarith).
generalize (Z_div_mod [|a|] [|b|] H0) (Z_div_pos [|a|] [|b|] H0).
unfold Z.div; destruct (Z.div_eucl [|a|] [|b|]).
rewrite ?phi_phi_inv.
destruct 1; intros.
rewrite H1, Z.mul_comm.
generalize (phi_bounded a)(phi_bounded b); intros.
replace (z0 mod wB) with z0 by (symmetry; apply Zmod_small; omega).
replace (z mod wB) with z; auto with zarith.
  symmetry; apply Zmod_small.
  split; auto with zarith.
  apply Z.le_lt_trans with [|a|]; auto with zarith.
  rewrite H1.
  apply Z.le_trans with ([|b|]*z); try omega.
  rewrite <- (Z.mul_1_l z) at 1.
  apply Z.mul_le_mono_nonneg; auto with zarith.
Qed.

Lemma spec_mod : forall a b, 0 < [|b |] ->
      [|let (_,r) := (a /b)%int31 in r |] = [|a |] mod [|b |].
Proof.
unfold div31; intros.
assert ([|b|]>0) by (auto with zarith).
unfold Z.modulo.
generalize (Z_div_mod [|a|] [|b|] H0).
destruct (Z.div_eucl [|a|] [|b|]).
rewrite ?phi_phi_inv.
destruct 1; intros.
generalize (phi_bounded b); intros.
apply Zmod_small; omega.
Qed.

Lemma phi_gcd : forall i j,
  [|gcd31 i j |] = Zgcdn (2*size) [|j |] [|i |].
Proof.
  unfold gcd31.
  induction (2*size)%nat; intros.
  reflexivity.
  simpl euler.
  unfold compare31.
  change [|On |] with 0.
  generalize (phi_bounded j)(phi_bounded i); intros.
  case_eq [|j|]; intros.
  simpl; intros.
  generalize (Zabs_spec [|i|]); omega.
  simpl. rewrite IHn, H1; f_equal.
  rewrite spec_mod, H1; auto.
  rewrite H1; compute; auto.
  rewrite H1 in H; destruct H as [H _]; compute in H; elim H; auto.
Qed.

Lemma spec_gcd : forall a b, Zis_gcd [|a |] [|b |] [|gcd31 a b |].
Proof.
intros.
rewrite phi_gcd.
apply Zis_gcd_sym.
apply Zgcdn_is_gcd.
unfold Zgcd_bound.
generalize (phi_bounded b).
destruct [|b|].
unfold size; auto with zarith.
intros (_,H).
cut (Pos.size_nat p <= size)%nat; [ omega | rewrite <- Zpower2_Psize; auto].
intros (H,_); compute in H; elim H; auto.
Qed.

Lemma iter_int31_iter_nat : forall A f i a,
  iter_int31 i A f a = iter_nat (Z.abs_nat [|i |]) A f a.
Proof.
intros.
unfold iter_int31.
rewrite <- recrbis_equiv; auto; unfold recrbis.
rewrite <- phibis_aux_equiv.

revert i a; induction size.
simpl; auto.
simpl; intros.
case_eq (firstr i); intros H; rewrite 2 IHn;
  unfold phibis_aux; simpl; rewrite ?H; fold (phibis_aux n (shiftr i));
  generalize (phibis_aux_pos n (shiftr i)); intros;
  set (z := phibis_aux n (shiftr i)) in *; clearbody z;
  rewrite <- nat_rect_plus.

f_equal.
rewrite Z.double_spec, <- Z.add_diag.
symmetry; apply Zabs2Nat.inj_add; auto with zarith.

change (iter_nat (S (Z.abs_nat z) + (Z.abs_nat z))%nat A f a =
    iter_nat (Z.abs_nat (Z.succ_double z)) A f a); f_equal.
rewrite Z.succ_double_spec, <- Z.add_diag.
rewrite Zabs2Nat.inj_add; auto with zarith.
rewrite Zabs2Nat.inj_add; auto with zarith.
change (Z.abs_nat 1) with 1%nat; omega.
Qed.

Fixpoint addmuldiv31_alt n i j :=
  match n with
    | O => i
    | S n => addmuldiv31_alt n (sneakl (firstl j) i) (shiftl j)
  end.

Lemma addmuldiv31_equiv : forall p x y,
  addmuldiv31 p x y = addmuldiv31_alt (Z.abs_nat [|p |]) x y.
Proof.
intros.
unfold addmuldiv31.
rewrite iter_int31_iter_nat.
set (n:=Z.abs_nat [|p|]); clearbody n; clear p.
revert x y; induction n.
simpl; auto.
intros.
simpl addmuldiv31_alt.
replace (S n) with (n+1)%nat by (rewrite plus_comm; auto).
rewrite nat_rect_plus; simpl; auto.
Qed.

Lemma spec_add_mul_div : forall x y p, [|p |] <= Zpos 31 ->
   [| addmuldiv31 p x y |] =
   ([|x |] * (2 ^ [|p |]) + [|y |] / (2 ^ ((Zpos 31) - [|p |]))) mod wB.
Proof.
intros.
rewrite addmuldiv31_equiv.
assert ([|p|] = Z.of_nat (Z.abs_nat [|p|])).
  rewrite Zabs2Nat.id_abs; symmetry; apply Z.abs_eq.
  destruct (phi_bounded p); auto.
rewrite H0; rewrite H0 in H; clear H0; rewrite Zabs2Nat.id.
set (n := Z.abs_nat [|p|]) in *; clearbody n.
assert (n <= 31)%nat.
  rewrite Nat2Z.inj_le; auto with zarith.
clear p H; revert x y.

induction n.
simpl Z.of_nat; intros.
rewrite Z.mul_1_r.
replace ([|y|] / 2^(31-0)) with 0.
rewrite Z.add_0_r.
symmetry; apply Zmod_small; apply phi_bounded.
symmetry; apply Zdiv_small; apply phi_bounded.

simpl addmuldiv31_alt; intros.
rewrite IHn; [ | omega ].
case_eq (firstl y); intros.

rewrite phi_twice, Z.double_spec.
rewrite phi_twice_firstl; auto.
change (Z.double [|y|]) with (2*[|y|]).
rewrite Nat2Z.inj_succ, Z.pow_succ_r; auto with zarith.
rewrite Zplus_mod; rewrite Zmult_mod_idemp_l; rewrite <- Zplus_mod.
f_equal.
f_equal.
ring.
replace (31-Z.of_nat n) with (Z.succ(31-Z.succ(Z.of_nat n))) by ring.
rewrite Z.pow_succ_r, <- Zdiv_Zdiv; auto with zarith.
rewrite Z.mul_comm, Z_div_mult; auto with zarith.

rewrite phi_twice_plus_one, Z.succ_double_spec.
rewrite phi_twice; auto.
change (Z.double [|y|]) with (2*[|y|]).
rewrite Nat2Z.inj_succ, Z.pow_succ_r; auto with zarith.
rewrite Zplus_mod; rewrite Zmult_mod_idemp_l; rewrite <- Zplus_mod.
rewrite Z.mul_add_distr_r, Z.mul_1_l, <- Z.add_assoc.
f_equal.
f_equal.
ring.
assert ((2*[|y|]) mod wB = 2*[|y|] - wB).
  clear - H. symmetry. apply Zmod_unique with 1; [ | ring ].
  generalize (phi_lowerbound _ H) (phi_bounded y).
  set (wB' := 2^Z.of_nat (pred size)).
  replace wB with (2*wB'); [ omega | ].
  unfold wB'. rewrite <- Z.pow_succ_r, <- Nat2Z.inj_succ by (auto with zarith).
  f_equal.
rewrite H1.
replace wB with (2^(Z.of_nat n)*2^(31-Z.of_nat n)) by
  (rewrite <- Zpower_exp; auto with zarith; f_equal; unfold size; ring).
unfold Z.sub; rewrite <- Z.mul_opp_l.
rewrite Z_div_plus; auto with zarith.
ring_simplify.
replace (31+-Z.of_nat n) with (Z.succ(31-Z.succ(Z.of_nat n))) by ring.
rewrite Z.pow_succ_r, <- Zdiv_Zdiv; auto with zarith.
rewrite Z.mul_comm, Z_div_mult; auto with zarith.
Qed.

Lemma shift_unshift_mod_2 : forall n p a, 0 <= p <= n ->
   ((a * 2 ^ (n - p )) mod (2^n ) / 2 ^ (n - p )) mod (2^n ) =
   a mod 2 ^ p.
Proof.
intros.
rewrite Zmod_small.
rewrite Zmod_eq by (auto with zarith).
unfold Z.sub at 1.
rewrite Z_div_plus_full_l
  by (cut (0 < 2^(n-p)); auto with zarith).
assert (2^n = 2^(n-p)*2^p).
  rewrite <- Zpower_exp by (auto with zarith).
  replace (n-p+p) with n; auto with zarith.
rewrite H0.
rewrite <- Zdiv_Zdiv, Z_div_mult by (auto with zarith).
rewrite (Z.mul_comm (2^(n-p))), Z.mul_assoc.
rewrite <- Z.mul_opp_l.
rewrite Z_div_mult by (auto with zarith).
symmetry; apply Zmod_eq; auto with zarith.

remember (a * 2 ^ (n - p)) as b.
destruct (Z_mod_lt b (2^n)); auto with zarith.
split.
apply Z_div_pos; auto with zarith.
apply Zdiv_lt_upper_bound; auto with zarith.
apply Z.lt_le_trans with (2^n); auto with zarith.
rewrite <- (Z.mul_1_r (2^n)) at 1.
apply Z.mul_le_mono_nonneg; auto with zarith.
cut (0 < 2 ^ (n-p)); auto with zarith.
Qed.

Lemma spec_pos_mod : forall w p,
       [|ZnZ.pos_mod p w |] = [|w |] mod (2 ^ [|p |]).
Proof.
unfold int31_ops, ZnZ.pos_mod, compare31.
change [|31|] with 31%Z.
assert (forall w p, 31<=p -> [|w |] = [|w |] mod 2^p).
  intros.
  generalize (phi_bounded w).
  symmetry; apply Zmod_small.
  split; auto with zarith.
  apply Z.lt_le_trans with wB; auto with zarith.
  apply Zpower_le_monotone; auto with zarith.
intros.
case_eq ([|p|] ?= 31); intros;
  [ apply H; rewrite (Z.compare_eq _ _ H0); auto with zarith | |
    apply H; change ([|p|]>31)%Z in H0; auto with zarith ].
change ([|p|]<31) in H0.
rewrite spec_add_mul_div by auto with zarith.
change [|0|] with 0%Z; rewrite Z.mul_0_l, Z.add_0_l.
generalize (phi_bounded p)(phi_bounded w); intros.
assert (31-[|p|]<wB).
  apply Z.le_lt_trans with 31%Z; auto with zarith.
  compute; auto.
assert ([|31-p|]=31-[|p|]).
  unfold sub31; rewrite phi_phi_inv.
  change [|31|] with 31%Z.
  apply Zmod_small; auto with zarith.
rewrite spec_add_mul_div by (rewrite H4; auto with zarith).
change [|0|] with 0%Z; rewrite Zdiv_0_l, Z.add_0_r.
rewrite H4.
apply shift_unshift_mod_2; simpl; auto with zarith.
Qed.

Shift operations

Lemma spec_head00: forall x, [|x |] = 0 -> [|head031 x |] = Zpos 31.
Proof.
  intros.
  generalize (phi_inv_phi x).
  rewrite H; simpl phi_inv.
  intros H'; rewrite <- H'.
  simpl; auto.
Qed.

Fixpoint head031_alt n x :=
  match n with
    | O => 0%nat
    | S n => match firstl x with
               | D0 => S (head031_alt n (shiftl x))
               | D1 => 0%nat
             end
  end.

Lemma head031_equiv :
   forall x, [|head031 x |] = Z.of_nat (head031_alt size x).
Proof.
intros.
case_eq (iszero x); intros.
rewrite (iszero_eq0 _ H).
simpl; auto.

unfold head031, recl.
change On with (phi_inv (Z.of_nat (31-size))).
replace (head031_alt size x) with
   (head031_alt size x + (31 - size ))%nat by auto.
assert (size <= 31)%nat by auto with arith.

revert x H; induction size; intros.
simpl; auto.
unfold recl_aux; fold recl_aux.
unfold head031_alt; fold head031_alt.
rewrite H.
assert ([|phi_inv (Z.of_nat (31-S n))|] = Z.of_nat (31 - S n)).
  rewrite phi_phi_inv.
  apply Zmod_small.
  split.
  change 0 with (Z.of_nat O); apply inj_le; omega.
  apply Z.le_lt_trans with (Z.of_nat 31).
  apply inj_le; omega.
  compute; auto.
case_eq (firstl x); intros; auto.
rewrite plus_Sn_m, plus_n_Sm.
replace (S (31 - S n)) with (31 - n)%nat by omega.
rewrite <- IHn; [ | omega | ].
f_equal; f_equal.
unfold add31.
rewrite H1.
f_equal.
change [|In |] with 1.
replace (31-n)%nat with (S (31 - S n))%nat by omega.
rewrite Nat2Z.inj_succ; ring.

clear - H H2.
rewrite (sneakr_shiftl x) in H.
rewrite H2 in H.
case_eq (iszero (shiftl x)); intros; auto.
rewrite (iszero_eq0 _ H0) in H; discriminate.
Qed.

Lemma phi_nz : forall x, 0 < [|x |] <-> x <> 0%int31.
Proof.
split; intros.
red; intro; subst x; discriminate.
assert ([|x|]<>0%Z).
contradict H.
rewrite <- (phi_inv_phi x); rewrite H; auto.
generalize (phi_bounded x); auto with zarith.
Qed.

Lemma spec_head0 : forall x, 0 < [|x |] ->
         wB / 2 <= 2 ^ ([|head031 x |]) * [|x |] < wB.
Proof.
intros.
rewrite head031_equiv.
assert (nshiftl x size = 0%int31).
  apply nshiftl_size.
revert x H H0.
unfold size at 2 5.
induction size.
simpl Z.of_nat.
intros.
compute in H0; rewrite H0 in H; discriminate.

intros.
simpl head031_alt.
case_eq (firstl x); intros.
rewrite (Nat2Z.inj_succ (head031_alt n (shiftl x))), Z.pow_succ_r; auto with zarith.
rewrite <- Z.mul_assoc, Z.mul_comm, <- Z.mul_assoc, <-(Z.mul_comm 2).
rewrite <- Z.double_spec, <- (phi_twice_firstl _ H1).
apply IHn.

rewrite phi_nz; rewrite phi_nz in H; contradict H.
change twice with shiftl in H.
rewrite (sneakr_shiftl x), H1, H; auto.

rewrite <- nshiftl_S_tail; auto.

change (2^(Z.of_nat 0)) with 1; rewrite Z.mul_1_l.
generalize (phi_bounded x); unfold size; split; auto with zarith.
change (2^(Z.of_nat 31)/2) with (2^(Z.of_nat (pred size))).
apply phi_lowerbound; auto.
Qed.

Lemma spec_tail00: forall x, [|x |] = 0 -> [|tail031 x |] = Zpos 31.
Proof.
  intros.
  generalize (phi_inv_phi x).
  rewrite H; simpl phi_inv.
  intros H'; rewrite <- H'.
  simpl; auto.
Qed.

Fixpoint tail031_alt n x :=
  match n with
    | O => 0%nat
    | S n => match firstr x with
               | D0 => S (tail031_alt n (shiftr x))
               | D1 => 0%nat
             end
  end.

Lemma tail031_equiv :
   forall x, [|tail031 x |] = Z.of_nat (tail031_alt size x).
Proof.
intros.
case_eq (iszero x); intros.
rewrite (iszero_eq0 _ H).
simpl; auto.

unfold tail031, recr.
change On with (phi_inv (Z.of_nat (31-size))).
replace (tail031_alt size x) with
   (tail031_alt size x + (31 - size ))%nat by auto.
assert (size <= 31)%nat by auto with arith.

revert x H; induction size; intros.
simpl; auto.
unfold recr_aux; fold recr_aux.
unfold tail031_alt; fold tail031_alt.
rewrite H.
assert ([|phi_inv (Z.of_nat (31-S n))|] = Z.of_nat (31 - S n)).
  rewrite phi_phi_inv.
  apply Zmod_small.
  split.
  change 0 with (Z.of_nat O); apply inj_le; omega.
  apply Z.le_lt_trans with (Z.of_nat 31).
  apply inj_le; omega.
  compute; auto.
case_eq (firstr x); intros; auto.
rewrite plus_Sn_m, plus_n_Sm.
replace (S (31 - S n)) with (31 - n)%nat by omega.
rewrite <- IHn; [ | omega | ].
f_equal; f_equal.
unfold add31.
rewrite H1.
f_equal.
change [|In |] with 1.
replace (31-n)%nat with (S (31 - S n))%nat by omega.
rewrite Nat2Z.inj_succ; ring.

clear - H H2.
rewrite (sneakl_shiftr x) in H.
rewrite H2 in H.
case_eq (iszero (shiftr x)); intros; auto.
rewrite (iszero_eq0 _ H0) in H; discriminate.
Qed.

Lemma spec_tail0 : forall x, 0 < [|x |] ->
         exists y, 0 <= y /\ [|x |] = (2 * y + 1) * (2 ^ [|tail031 x |]).
Proof.
intros.
rewrite tail031_equiv.
assert (nshiftr x size = 0%int31).
  apply nshiftr_size.
revert x H H0.
induction size.
simpl Z.of_nat.
intros.
compute in H0; rewrite H0 in H; discriminate.

intros.
simpl tail031_alt.
case_eq (firstr x); intros.
rewrite (Nat2Z.inj_succ (tail031_alt n (shiftr x))), Z.pow_succ_r; auto with zarith.
destruct (IHn (shiftr x)) as (y & Hy1 & Hy2).

rewrite phi_nz; rewrite phi_nz in H; contradict H.
rewrite (sneakl_shiftr x), H1, H; auto.

rewrite <- nshiftr_S_tail; auto.

exists y; split; auto.
rewrite phi_eqn1; auto.
rewrite Z.double_spec, Hy2; ring.

exists [|shiftr x|].
split.
generalize (phi_bounded (shiftr x)); auto with zarith.
rewrite phi_eqn2; auto.
rewrite Z.succ_double_spec; simpl; ring.
Qed.

Lemma quotient_by_2 a: a - 1 <= (a /2) + (a /2).
Proof.
case (Z_mod_lt a 2); auto with zarith.
intros H1; rewrite Zmod_eq_full; auto with zarith.
Qed.

Lemma sqrt_main_trick j k: 0 <= j -> 0 <= k ->
   (j * k ) + j <= ((j + k )/2 + 1) ^ 2.
Proof.
intros Hj; generalize Hj k; pattern j; apply natlike_ind;
   auto; clear k j Hj.
intros _ k Hk; repeat rewrite Z.add_0_l.
apply Z.mul_nonneg_nonneg; generalize (Z_div_pos k 2); auto with zarith.
intros j Hj Hrec _ k Hk; pattern k; apply natlike_ind; auto; clear k Hk.
rewrite Z.mul_0_r, Z.add_0_r, Z.add_0_l.
generalize (sqr_pos (Z.succ j / 2)) (quotient_by_2 (Z.succ j));
   unfold Z.succ.
rewrite Z.pow_2_r, Z.mul_add_distr_r; repeat rewrite Z.mul_add_distr_l.
auto with zarith.
intros k Hk _.
replace ((Z.succ j + Z.succ k) / 2) with ((j + k)/2 + 1).
generalize (Hrec Hj k Hk) (quotient_by_2 (j + k)).
unfold Z.succ; repeat rewrite Z.pow_2_r;
   repeat rewrite Z.mul_add_distr_r; repeat rewrite Z.mul_add_distr_l.
repeat rewrite Z.mul_1_l; repeat rewrite Z.mul_1_r.
auto with zarith.
rewrite Z.add_comm, <- Z_div_plus_full_l; auto with zarith.
apply f_equal2 with (f := Z.div); auto with zarith.
Qed.

Lemma sqrt_main i j: 0 <= i -> 0 < j -> i < ((j + (i /j ))/2 + 1) ^ 2.
Proof.
intros Hi Hj.
assert (Hij: 0 <= i/j) by (apply Z_div_pos; auto with zarith).
apply Z.lt_le_trans with (2 := sqrt_main_trick _ _ (Z.lt_le_incl _ _ Hj) Hij).
pattern i at 1; rewrite (Z_div_mod_eq i j); case (Z_mod_lt i j); auto with zarith.
Qed.

Lemma sqrt_init i: 1 < i -> i < (i /2 + 1) ^ 2.
Proof.
intros Hi.
assert (H1: 0 <= i - 2) by auto with zarith.
assert (H2: 1 <= (i / 2) ^ 2); auto with zarith.
   replace i with (1* 2 + (i - 2)); auto with zarith.
   rewrite Z.pow_2_r, Z_div_plus_full_l; auto with zarith.
   generalize (sqr_pos ((i - 2)/ 2)) (Z_div_pos (i - 2) 2).
   rewrite Z.mul_add_distr_r; repeat rewrite Z.mul_add_distr_l.
   auto with zarith.
generalize (quotient_by_2 i).
rewrite Z.pow_2_r in H2 |- *;
   repeat (rewrite Z.mul_add_distr_r ||
           rewrite Z.mul_add_distr_l ||
           rewrite Z.mul_1_l || rewrite Z.mul_1_r).
   auto with zarith.
Qed.

Lemma sqrt_test_true i j: 0 <= i -> 0 < j -> i /j >= j -> j ^ 2 <= i.
Proof.
intros Hi Hj Hd; rewrite Z.pow_2_r.
apply Z.le_trans with (j * (i/j)); auto with zarith.
apply Z_mult_div_ge; auto with zarith.
Qed.

Lemma sqrt_test_false i j: 0 <= i -> 0 < j -> i /j < j -> (j + (i /j ))/2 < j.
Proof.
intros Hi Hj H; case (Z.le_gt_cases j ((j + (i/j))/2)); auto.
intros H1; contradict H; apply Z.le_ngt.
assert (2 * j <= j + (i/j)); auto with zarith.
apply Z.le_trans with (2 * ((j + (i/j))/2)); auto with zarith.
apply Z_mult_div_ge; auto with zarith.
Qed.

Lemma sqrt31_step_def rec i j:
   sqrt31_step rec i j =
   match (fst (i /j) ?= j)%int31 with
     Lt => rec i (fst ((j + fst(i /j))/2))%int31
   | _ => j
   end.
Proof.
unfold sqrt31_step; case div31; intros.
simpl; case compare31; auto.
Qed.

Lemma div31_phi i j: 0 < [|j |] -> [|fst (i /j)%int31|] = [|i |]/[|j |].
intros Hj; generalize (spec_div i j Hj).
case div31; intros q r; simpl @fst.
intros (H1,H2); apply Zdiv_unique with [|r|]; auto with zarith.
rewrite H1; ring.
Qed.

Lemma sqrt31_step_correct rec i j:
  0 < [|i |] -> 0 < [|j |] -> [|i |] < ([|j |] + 1) ^ 2 ->
   2 * [|j |] < wB ->
  (forall j1 : int31,
    0 < [|j1 |] < [|j |] -> [|i |] < ([|j1 |] + 1) ^ 2 ->
    [|rec i j1 |] ^ 2 <= [|i |] < ([|rec i j1 |] + 1) ^ 2) ->
  [|sqrt31_step rec i j |] ^ 2 <= [|i |] < ([|sqrt31_step rec i j |] + 1) ^ 2.
Proof.
assert (Hp2: 0 < [|2|]) by exact (eq_refl Lt).
intros Hi Hj Hij H31 Hrec; rewrite sqrt31_step_def.
rewrite spec_compare, div31_phi; auto.
case Z.compare_spec; auto; intros Hc;
   try (split; auto; apply sqrt_test_true; auto with zarith; fail).
assert (E : [|(j + fst (i / j)%int31)|] = [|j|] + [|i|] / [|j|]).
{ rewrite spec_add, div31_phi; auto using Z.mod_small with zarith. }
apply Hrec; rewrite !div31_phi, E; auto using sqrt_main with zarith.
split; try apply sqrt_test_false; auto with zarith.
apply Z.le_succ_l in Hj. change (1 <= [|j|]) in Hj.
Z.le_elim Hj.
- replace ([|j|] + [|i|]/[|j|]) with
   (1 * 2 + (([|j|] - 2) + [|i|] / [|j|])) by ring.
   rewrite Z_div_plus_full_l; auto with zarith.
   assert (0 <= [|i|]/ [|j|]) by auto with zarith.
   assert (0 <= ([|j|] - 2 + [|i|] / [|j|]) / [|2|]); auto with zarith.
- rewrite <- Hj, Zdiv_1_r.
   replace (1 + [|i|]) with (1 * 2 + ([|i|] - 1)) by ring.
   rewrite Z_div_plus_full_l; auto with zarith.
   assert (0 <= ([|i|] - 1) /2) by auto with zarith.
   change ([|2|]) with 2; auto with zarith.
Qed.

Lemma iter31_sqrt_correct n rec i j: 0 < [|i |] -> 0 < [|j |] ->
  [|i |] < ([|j |] + 1) ^ 2 -> 2 * [|j |] < 2 ^ (Z.of_nat size ) ->
  (forall j1, 0 < [|j1 |] -> 2^(Z.of_nat n ) + [|j1 |] <= [|j |] ->
      [|i |] < ([|j1 |] + 1) ^ 2 -> 2 * [|j1 |] < 2 ^ (Z.of_nat size ) ->
       [|rec i j1 |] ^ 2 <= [|i |] < ([|rec i j1 |] + 1) ^ 2) ->
  [|iter31_sqrt n rec i j |] ^ 2 <= [|i |] < ([|iter31_sqrt n rec i j |] + 1) ^ 2.
Proof.
revert rec i j; elim n; unfold iter31_sqrt; fold iter31_sqrt; clear n.
intros rec i j Hi Hj Hij H31 Hrec; apply sqrt31_step_correct; auto with zarith.
intros; apply Hrec; auto with zarith.
rewrite Z.pow_0_r; auto with zarith.
intros n Hrec rec i j Hi Hj Hij H31 HHrec.
apply sqrt31_step_correct; auto.
intros j1 Hj1 Hjp1; apply Hrec; auto with zarith.
intros j2 Hj2 H2j2 Hjp2 Hj31; apply Hrec; auto with zarith.
intros j3 Hj3 Hpj3.
apply HHrec; auto.
rewrite Nat2Z.inj_succ, Z.pow_succ_r.
apply Z.le_trans with (2 ^Z.of_nat n + [|j2|]); auto with zarith.
apply Nat2Z.is_nonneg.
Qed.

Lemma spec_sqrt : forall x,
       [|sqrt31 x |] ^ 2 <= [|x |] < ([|sqrt31 x |] + 1) ^ 2.
Proof.
intros i; unfold sqrt31.
rewrite spec_compare. case Z.compare_spec; change [|1|] with 1;
   intros Hi; auto with zarith.
repeat rewrite Z.pow_2_r; auto with zarith.
apply iter31_sqrt_correct; auto with zarith.
  rewrite div31_phi; change ([|2|]) with 2; auto with zarith.
  replace ([|i|]) with (1 * 2 + ([|i|] - 2))%Z; try ring.
  assert (0 <= ([|i|] - 2)/2)%Z by (apply Z_div_pos; auto with zarith).
  rewrite Z_div_plus_full_l; auto with zarith.
  rewrite div31_phi; change ([|2|]) with 2; auto with zarith.
  apply sqrt_init; auto.
rewrite div31_phi; change ([|2|]) with 2; auto with zarith.
apply Z.le_lt_trans with ([|i|]).
apply Z_mult_div_ge; auto with zarith.
case (phi_bounded i); auto.
intros j2 H1 H2; contradict H2; apply Z.lt_nge.
rewrite div31_phi; change ([|2|]) with 2; auto with zarith.
apply Z.le_lt_trans with ([|i|]); auto with zarith.
assert (0 <= [|i|]/2)%Z by (apply Z_div_pos; auto with zarith).
apply Z.le_trans with (2 * ([|i|]/2)); auto with zarith.
apply Z_mult_div_ge; auto with zarith.
case (phi_bounded i); unfold size; auto with zarith.
change [|0|] with 0; auto with zarith.
case (phi_bounded i); repeat rewrite Z.pow_2_r; auto with zarith.
Qed.

Lemma sqrt312_step_def rec ih il j:
   sqrt312_step rec ih il j =
   match (ih ?= j)%int31 with
      Eq => j
    | Gt => j
    | _ =>
       match (fst (div3121 ih il j) ?= j)%int31 with
         Lt => let m := match j +c fst (div3121 ih il j) with
                       C0 m1 => fst (m1/2)%int31
                     | C1 m1 => (fst (m1/2) + v30)%int31
                     end in rec ih il m
       | _ => j
       end
   end.
Proof.
unfold sqrt312_step; case div3121; intros.
simpl; case compare31; auto.
Qed.

Lemma sqrt312_lower_bound ih il j:
   phi2 ih il < ([|j |] + 1) ^ 2 -> [|ih |] <= [|j |].
Proof.
intros H1.
case (phi_bounded j); intros Hbj _.
case (phi_bounded il); intros Hbil _.
case (phi_bounded ih); intros Hbih Hbih1.
assert ([|ih|] < [|j|] + 1); auto with zarith.
apply Z.square_lt_simpl_nonneg; auto with zarith.
rewrite <- ?Z.pow_2_r; apply Z.le_lt_trans with (2 := H1).
apply Z.le_trans with ([|ih|] * wB).
- rewrite ? Z.pow_2_r; auto with zarith.
- unfold phi2. change base with wB; auto with zarith.
Qed.

Lemma div312_phi ih il j: (2^30 <= [|j |] -> [|ih |] < [|j |] ->
  [|fst (div3121 ih il j)|] = phi2 ih il /[|j |])%Z.
Proof.
intros Hj Hj1.
generalize (spec_div21 ih il j Hj Hj1).
case div3121; intros q r (Hq, Hr).
apply Zdiv_unique with (phi r); auto with zarith.
simpl @fst; apply eq_trans with (1 := Hq); ring.
Qed.

Lemma sqrt312_step_correct rec ih il j:
  2 ^ 29 <= [|ih |] -> 0 < [|j |] -> phi2 ih il < ([|j |] + 1) ^ 2 ->
  (forall j1, 0 < [|j1 |] < [|j |] -> phi2 ih il < ([|j1 |] + 1) ^ 2 ->
     [|rec ih il j1 |] ^ 2 <= phi2 ih il < ([|rec ih il j1 |] + 1) ^ 2) ->
  [|sqrt312_step rec ih il j |] ^ 2 <= phi2 ih il
      < ([|sqrt312_step rec ih il j |] + 1) ^ 2.
Proof.
assert (Hp2: (0 < [|2|])%Z) by exact (eq_refl Lt).
intros Hih Hj Hij Hrec; rewrite sqrt312_step_def.
assert (H1: ([|ih|] <= [|j|])) by (apply sqrt312_lower_bound with il; auto).
case (phi_bounded ih); intros Hih1 _.
case (phi_bounded il); intros Hil1 _.
case (phi_bounded j); intros _ Hj1.
assert (Hp3: (0 < phi2 ih il)).
{ unfold phi2; apply Z.lt_le_trans with ([|ih|] * base); auto with zarith.
   apply Z.mul_pos_pos; auto with zarith.
   apply Z.lt_le_trans with (2:= Hih); auto with zarith. }
rewrite spec_compare. case Z.compare_spec; intros Hc1.
- split; auto.
   apply sqrt_test_true; auto.
   + unfold phi2, base; auto with zarith.
   + unfold phi2; rewrite Hc1.
     assert (0 <= [|il|]/[|j|]) by (apply Z_div_pos; auto with zarith).
     rewrite Z.mul_comm, Z_div_plus_full_l; auto with zarith.
     change base with wB. auto with zarith.
- case (Z.le_gt_cases (2 ^ 30) [|j|]); intros Hjj.
   + rewrite spec_compare; case Z.compare_spec;
     rewrite div312_phi; auto; intros Hc;
     try (split; auto; apply sqrt_test_true; auto with zarith; fail).
     apply Hrec.
     * assert (Hf1: 0 <= phi2 ih il/ [|j|]) by auto with zarith.
       apply Z.le_succ_l in Hj. change (1 <= [|j|]) in Hj.
       Z.le_elim Hj;
         [ | contradict Hc; apply Z.le_ngt;
             rewrite <- Hj, Zdiv_1_r; auto with zarith ].
       assert (Hf3: 0 < ([|j|] + phi2 ih il / [|j|]) / 2).
       { replace ([|j|] + phi2 ih il/ [|j|]) with
         (1 * 2 + (([|j|] - 2) + phi2 ih il / [|j|])); try ring.
         rewrite Z_div_plus_full_l; auto with zarith.
         assert (0 <= ([|j|] - 2 + phi2 ih il / [|j|]) / 2) ;
         auto with zarith. }
       assert (Hf4: ([|j|] + phi2 ih il / [|j|]) / 2 < [|j|]).
       { apply sqrt_test_false; auto with zarith. }
       generalize (spec_add_c j (fst (div3121 ih il j))).
       unfold interp_carry; case add31c; intros r;
       rewrite div312_phi; auto with zarith.
       { rewrite div31_phi; change [|2|] with 2; auto with zarith.
         intros HH; rewrite HH; clear HH; auto with zarith. }
       { rewrite spec_add, div31_phi; change [|2|] with 2; auto.
         rewrite Z.mul_1_l; intros HH.
         rewrite Z.add_comm, <- Z_div_plus_full_l; auto with zarith.
         change (phi v30 * 2) with (2 ^ Z.of_nat size).
         rewrite HH, Zmod_small; auto with zarith. }
     * replace (phi _) with (([|j|] + (phi2 ih il)/([|j|]))/2);
       [ apply sqrt_main; auto with zarith | ].
       generalize (spec_add_c j (fst (div3121 ih il j))).
       unfold interp_carry; case add31c; intros r;
       rewrite div312_phi; auto with zarith.
       { rewrite div31_phi; auto with zarith.
         intros HH; rewrite HH; auto with zarith. }
       { intros HH; rewrite <- HH.
         change (1 * 2 ^ Z.of_nat size) with (phi (v30) * 2).
         rewrite Z_div_plus_full_l; auto with zarith.
         rewrite Z.add_comm.
         rewrite spec_add, Zmod_small.
         - rewrite div31_phi; auto.
         - split; auto with zarith.
           + case (phi_bounded (fst (r/2)%int31));
             case (phi_bounded v30); auto with zarith.
           + rewrite div31_phi; change (phi 2) with 2; auto.
             change (2 ^Z.of_nat size) with (base /2 + phi v30).
             assert (phi r / 2 < base /2); auto with zarith.
             apply Z.mul_lt_mono_pos_r with 2; auto with zarith.
             change (base /2 * 2) with base.
             apply Z.le_lt_trans with (phi r).
             * rewrite Z.mul_comm; apply Z_mult_div_ge; auto with zarith.
             * case (phi_bounded r); auto with zarith. }
   + contradict Hij; apply Z.le_ngt.
     assert ((1 + [|j|]) <= 2 ^ 30); auto with zarith.
     apply Z.le_trans with ((2 ^ 30) * (2 ^ 30)); auto with zarith.
     * assert (0 <= 1 + [|j|]); auto with zarith.
       apply Z.mul_le_mono_nonneg; auto with zarith.
     * change ((2 ^ 30) * (2 ^ 30)) with ((2 ^ 29) * base).
       apply Z.le_trans with ([|ih|] * base);
         change wB with base in *; auto with zarith.
       unfold phi2, base; auto with zarith.
- split; auto.
   apply sqrt_test_true; auto.
   + unfold phi2, base; auto with zarith.
   + apply Z.le_ge; apply Z.le_trans with (([|j|] * base )/[|j|]).
     * rewrite Z.mul_comm, Z_div_mult; auto with zarith.
     * apply Z.ge_le; apply Z_div_ge; auto with zarith.
Qed.

Lemma iter312_sqrt_correct n rec ih il j:
  2^29 <= [|ih |] -> 0 < [|j |] -> phi2 ih il < ([|j |] + 1) ^ 2 ->
  (forall j1, 0 < [|j1 |] -> 2^(Z.of_nat n ) + [|j1 |] <= [|j |] ->
      phi2 ih il < ([|j1 |] + 1) ^ 2 ->
       [|rec ih il j1 |] ^ 2 <= phi2 ih il < ([|rec ih il j1 |] + 1) ^ 2) ->
  [|iter312_sqrt n rec ih il j |] ^ 2 <= phi2 ih il
      < ([|iter312_sqrt n rec ih il j |] + 1) ^ 2.
Proof.
revert rec ih il j; elim n; unfold iter312_sqrt; fold iter312_sqrt; clear n.
intros rec ih il j Hi Hj Hij Hrec; apply sqrt312_step_correct; auto with zarith.
intros; apply Hrec; auto with zarith.
rewrite Z.pow_0_r; auto with zarith.
intros n Hrec rec ih il j Hi Hj Hij HHrec.
apply sqrt312_step_correct; auto.
intros j1 Hj1 Hjp1; apply Hrec; auto with zarith.
intros j2 Hj2 H2j2 Hjp2; apply Hrec; auto with zarith.
intros j3 Hj3 Hpj3.
apply HHrec; auto.
rewrite Nat2Z.inj_succ, Z.pow_succ_r.
apply Z.le_trans with (2 ^Z.of_nat n + [|j2|]); auto with zarith.
apply Nat2Z.is_nonneg.
Qed.

Strategy 1 [iter312_sqrt].

Lemma spec_sqrt2 : forall x y,
       wB / 4 <= [|x |] ->
       let (s,r) := sqrt312 x y in
          [||WW x y ||] = [|s |] ^ 2 + [+|r |] /\
          [+|r |] <= 2 * [|s |].
Proof.
intros ih il Hih; unfold sqrt312.
change [||WW ih il||] with (phi2 ih il).
assert (Hbin: forall s, s * s + 2* s + 1 = (s + 1) ^ 2) by
  (intros s; ring).
assert (Hb: 0 <= base) by (red; intros HH; discriminate).
assert (Hi2: phi2 ih il < (phi Tn + 1) ^ 2).
{ change ((phi Tn + 1) ^ 2) with (2^62).
   apply Z.le_lt_trans with ((2^31 -1) * base + (2^31 - 1)); auto with zarith.
   2: simpl; unfold Z.pow_pos; simpl; auto with zarith.
   case (phi_bounded ih); case (phi_bounded il); intros H1 H2 H3 H4.
   unfold base, Z.pow, Z.pow_pos in H2,H4; simpl in H2,H4.
   unfold phi2. cbn [Z.pow Z.pow_pos Pos.iter]. auto with zarith. }
case (iter312_sqrt_correct 31 (fun _ _ j => j) ih il Tn); auto with zarith.
change [|Tn |] with 2147483647; auto with zarith.
intros j1 _ HH; contradict HH.
apply Z.lt_nge.
change [|Tn |] with 2147483647; auto with zarith.
change (2 ^ Z.of_nat 31) with 2147483648; auto with zarith.
case (phi_bounded j1); auto with zarith.
set (s := iter312_sqrt 31 (fun _ _ j : int31 => j) ih il Tn).
intros Hs1 Hs2.
generalize (spec_mul_c s s); case mul31c.
simpl zn2z_to_Z; intros HH.
assert ([|s|] = 0).
{ symmetry in HH. rewrite Z.mul_eq_0 in HH. destruct HH; auto. }
contradict Hs2; apply Z.le_ngt; rewrite H.
change ((0 + 1) ^ 2) with 1.
apply Z.le_trans with (2 ^ Z.of_nat size / 4 * base).
simpl; auto with zarith.
apply Z.le_trans with ([|ih|] * base); auto with zarith.
unfold phi2; case (phi_bounded il); auto with zarith.
intros ih1 il1.
change [||WW ih1 il1||] with (phi2 ih1 il1).
intros Hihl1.
generalize (spec_sub_c il il1).
case sub31c; intros il2 Hil2.
rewrite spec_compare; case Z.compare_spec.
unfold interp_carry in *.
intros H1; split.
rewrite Z.pow_2_r, <- Hihl1.
unfold phi2; ring[Hil2 H1].
replace [|il2|] with (phi2 ih il - phi2 ih1 il1).
rewrite Hihl1.
rewrite <-Hbin in Hs2; auto with zarith.
unfold phi2; rewrite H1, Hil2; ring.
unfold interp_carry.
intros H1; contradict Hs1.
apply Z.lt_nge; rewrite Z.pow_2_r, <-Hihl1.
unfold phi2.
case (phi_bounded il); intros _ H2.
apply Z.lt_le_trans with (([|ih|] + 1) * base + 0).
rewrite Z.mul_add_distr_r, Z.add_0_r; auto with zarith.
case (phi_bounded il1); intros H3 _.
apply Z.add_le_mono; auto with zarith.
unfold interp_carry in *; change (1 * 2 ^ Z.of_nat size) with base.
rewrite Z.pow_2_r, <- Hihl1, Hil2.
intros H1.
rewrite <- Z.le_succ_l, <- Z.add_1_r in H1.
Z.le_elim H1.
contradict Hs2; apply Z.le_ngt.
replace (([|s|] + 1) ^ 2) with (phi2 ih1 il1 + 2 * [|s|] + 1).
unfold phi2.
case (phi_bounded il); intros Hpil _.
assert (Hl1l: [|il1|] <= [|il|]).
{ case (phi_bounded il2); rewrite Hil2; auto with zarith. }
assert ([|ih1|] * base + 2 * [|s|] + 1 <= [|ih|] * base); auto with zarith.
case (phi_bounded s); change (2 ^ Z.of_nat size) with base; intros _ Hps.
case (phi_bounded ih1); intros Hpih1 _; auto with zarith.
apply Z.le_trans with (([|ih1|] + 2) * base); auto with zarith.
rewrite Z.mul_add_distr_r.
assert (2 * [|s|] + 1 <= 2 * base); auto with zarith.
rewrite Hihl1, Hbin; auto.
split.
unfold phi2; rewrite <- H1; ring.
replace (base + ([|il|] - [|il1|])) with (phi2 ih il - ([|s|] * [|s|])).
rewrite <-Hbin in Hs2; auto with zarith.
rewrite <- Hihl1; unfold phi2; rewrite <- H1; ring.
unfold interp_carry in Hil2 |- *.
unfold interp_carry; change (1 * 2 ^ Z.of_nat size) with base.
assert (Hsih: [|ih - 1|] = [|ih|] - 1).
{ rewrite spec_sub, Zmod_small; auto; change [|1|] with 1.
   case (phi_bounded ih); intros H1 H2.
   generalize Hih; change (2 ^ Z.of_nat size / 4) with 536870912.
   split; auto with zarith. }
rewrite spec_compare; case Z.compare_spec.
rewrite Hsih.
intros H1; split.
rewrite Z.pow_2_r, <- Hihl1.
unfold phi2; rewrite <-H1.
transitivity ([|ih|] * base + [|il1|] + ([|il|] - [|il1|])).
ring.
rewrite <-Hil2.
change (2 ^ Z.of_nat size) with base; ring.
replace [|il2|] with (phi2 ih il - phi2 ih1 il1).
rewrite Hihl1.
rewrite <-Hbin in Hs2; auto with zarith.
unfold phi2.
rewrite <-H1.
ring_simplify.
transitivity (base + ([|il|] - [|il1|])).
ring.
rewrite <-Hil2.
change (2 ^ Z.of_nat size) with base; ring.
rewrite Hsih; intros H1.
assert (He: [|ih|] = [|ih1|]).
{ apply Z.le_antisymm; auto with zarith.
   case (Z.le_gt_cases [|ih1|] [|ih|]); auto; intros H2.
   contradict Hs1; apply Z.lt_nge; rewrite Z.pow_2_r, <-Hihl1.
   unfold phi2.
   case (phi_bounded il); change (2 ^ Z.of_nat size) with base;
    intros _ Hpil1.
   apply Z.lt_le_trans with (([|ih|] + 1) * base).
   rewrite Z.mul_add_distr_r, Z.mul_1_l; auto with zarith.
   case (phi_bounded il1); intros Hpil2 _.
   apply Z.le_trans with (([|ih1|]) * base); auto with zarith. }
rewrite Z.pow_2_r, <-Hihl1; unfold phi2; rewrite <-He.
contradict Hs1; apply Z.lt_nge; rewrite Z.pow_2_r, <-Hihl1.
unfold phi2; rewrite He.
assert (phi il - phi il1 < 0); auto with zarith.
rewrite <-Hil2.
case (phi_bounded il2); auto with zarith.
intros H1.
rewrite Z.pow_2_r, <-Hihl1.
assert (H2 : [|ih1|]+2 <= [|ih|]); auto with zarith.
Z.le_elim H2.
contradict Hs2; apply Z.le_ngt.
replace (([|s|] + 1) ^ 2) with (phi2 ih1 il1 + 2 * [|s|] + 1).
unfold phi2.
assert ([|ih1|] * base + 2 * phi s + 1 <= [|ih|] * base + ([|il|] - [|il1|]));
  auto with zarith.
rewrite <-Hil2.
change (-1 * 2 ^ Z.of_nat size) with (-base).
case (phi_bounded il2); intros Hpil2 _.
apply Z.le_trans with ([|ih|] * base + - base); auto with zarith.
case (phi_bounded s); change (2 ^ Z.of_nat size) with base; intros _ Hps.
assert (2 * [|s|] + 1 <= 2 * base); auto with zarith.
apply Z.le_trans with ([|ih1|] * base + 2 * base); auto with zarith.
assert (Hi: ([|ih1|] + 3) * base <= [|ih|] * base); auto with zarith.
rewrite Z.mul_add_distr_r in Hi; auto with zarith.
rewrite Hihl1, Hbin; auto.
unfold phi2; rewrite <-H2.
split.
replace [|il|] with (([|il|] - [|il1|]) + [|il1|]); try ring.
rewrite <-Hil2.
change (-1 * 2 ^ Z.of_nat size) with (-base); ring.
replace (base + [|il2|]) with (phi2 ih il - phi2 ih1 il1).
rewrite Hihl1.
rewrite <-Hbin in Hs2; auto with zarith.
unfold phi2; rewrite <-H2.
replace [|il|] with (([|il|] - [|il1|]) + [|il1|]); try ring.
rewrite <-Hil2.
change (-1 * 2 ^ Z.of_nat size) with (-base); ring.
Qed.

iszero

Lemma spec_eq0 : forall x, ZnZ.eq0 x = true -> [|x |] = 0.
Proof.
clear; unfold ZnZ.eq0, int31_ops.
unfold compare31; intros.
change [|0|] with 0 in H.
apply Z.compare_eq.
now destruct ([|x|] ?= 0).
Qed.

Lemma spec_is_even : forall x,
      if ZnZ.is_even x then [|x |] mod 2 = 0 else [|x |] mod 2 = 1.
Proof.
unfold ZnZ.is_even, int31_ops; intros.
generalize (spec_div x 2).
destruct (x/2)%int31 as (q,r); intros.
unfold compare31.
change [|2|] with 2 in H.
change [|0|] with 0.
destruct H; auto with zarith.
replace ([|x|] mod 2) with [|r|].
destruct H; auto with zarith.
case Z.compare_spec; auto with zarith.
apply Zmod_unique with [|q|]; auto with zarith.
Qed.

Lemma log2_phi_bounded x : Z.log2 [|x |] < Z.of_nat size.
Proof.
  destruct (phi_bounded x) as (H,H').
  Z.le_elim H.
  - now apply Z.log2_lt_pow2.
  - now rewrite <- H.
Qed.

Lemma spec_lor x y : [| ZnZ.lor x y |] = Z.lor [|x |] [|y |].
Proof.
unfold ZnZ.lor,int31_ops. unfold lor31.
rewrite phi_phi_inv.
apply Z.mod_small; split; trivial.
- apply Z.lor_nonneg; split; apply phi_bounded.
- apply Z.log2_lt_cancel. rewrite Z.log2_pow2 by easy.
   rewrite Z.log2_lor; try apply phi_bounded.
   apply Z.max_lub_lt; apply log2_phi_bounded.
Qed.

Lemma spec_land x y : [| ZnZ.land x y |] = Z.land [|x |] [|y |].
Proof.
unfold ZnZ.land, int31_ops. unfold land31.
rewrite phi_phi_inv.
apply Z.mod_small; split; trivial.
- apply Z.land_nonneg; left; apply phi_bounded.
- apply Z.log2_lt_cancel. rewrite Z.log2_pow2 by easy.
   eapply Z.le_lt_trans.
   apply Z.log2_land; try apply phi_bounded.
   apply Z.min_lt_iff; left; apply log2_phi_bounded.
Qed.

Lemma spec_lxor x y : [| ZnZ.lxor x y |] = Z.lxor [|x |] [|y |].
Proof.
unfold ZnZ.lxor, int31_ops. unfold lxor31.
rewrite phi_phi_inv.
apply Z.mod_small; split; trivial.
- apply Z.lxor_nonneg; split; intros; apply phi_bounded.
- apply Z.log2_lt_cancel. rewrite Z.log2_pow2 by easy.
   eapply Z.le_lt_trans.
   apply Z.log2_lxor; try apply phi_bounded.
   apply Z.max_lub_lt; apply log2_phi_bounded.
Qed.

Global Instance int31_specs : ZnZ.Specs int31_ops := {
    spec_to_Z := phi_bounded;
    spec_of_pos := positive_to_int31_spec;
    spec_zdigits := spec_zdigits;
    spec_more_than_1_digit := spec_more_than_1_digit;
    spec_0 := spec_0;
    spec_1 := spec_1;
    spec_m1 := spec_m1;
    spec_compare := spec_compare;
    spec_eq0 := spec_eq0;
    spec_opp_c := spec_opp_c;
    spec_opp := spec_opp;
    spec_opp_carry := spec_opp_carry;
    spec_succ_c := spec_succ_c;
    spec_add_c := spec_add_c;
    spec_add_carry_c := spec_add_carry_c;
    spec_succ := spec_succ;
    spec_add := spec_add;
    spec_add_carry := spec_add_carry;
    spec_pred_c := spec_pred_c;
    spec_sub_c := spec_sub_c;
    spec_sub_carry_c := spec_sub_carry_c;
    spec_pred := spec_pred;
    spec_sub := spec_sub;
    spec_sub_carry := spec_sub_carry;
    spec_mul_c := spec_mul_c;
    spec_mul := spec_mul;
    spec_square_c := spec_square_c;
    spec_div21 := spec_div21;
    spec_div_gt := fun a b _ => spec_div a b;
    spec_div := spec_div;
    spec_modulo_gt := fun a b _ => spec_mod a b;
    spec_modulo := spec_mod;
    spec_gcd_gt := fun a b _ => spec_gcd a b;
    spec_gcd := spec_gcd;
    spec_head00 := spec_head00;
    spec_head0 := spec_head0;
    spec_tail00 := spec_tail00;
    spec_tail0 := spec_tail0;
    spec_add_mul_div := spec_add_mul_div;
    spec_pos_mod := spec_pos_mod;
    spec_is_even := spec_is_even;
    spec_sqrt2 := spec_sqrt2;
    spec_sqrt := spec_sqrt;
    spec_lor := spec_lor;
    spec_land := spec_land;
    spec_lxor := spec_lxor }.

End Int31_Specs.

Module Int31Cyclic <: CyclicType.
  Definition t := int31.
  Definition ops := int31_ops.
  Definition specs := int31_specs.
End Int31Cyclic.