Remove the omega tactic and related options

12 files changed, 17 insertions, 580 deletions

diff --git a/theories/Compat/Coq813.v b/theories/Compat/Coq813.v
index fe7431dcd3..5cfb9d84c7 100644
--- a/theories/Compat/Coq813.v
+++ b/theories/Compat/Coq813.v
@@ -11,3 +11,13 @@
 (** Compatibility file for making Coq act similar to Coq v8.13 *)
 
 Require Export Coq.Compat.Coq814.
+
+Require Coq.micromega.Lia.
+Module Export Coq.
+  Module Export omega.
+    Module Export Omega.
+      #[deprecated(since="8.12", note="The omega tactic was removed in v8.14.  You're now relying on the lia tactic.")]
+      Ltac omega := Lia.lia.
+    End Omega.
+  End omega.
+End Coq.
diff --git a/theories/ZArith/BinInt.v b/theories/ZArith/BinInt.v
index 47137414dc..ff1a2b5a53 100644
--- a/theories/ZArith/BinInt.v
+++ b/theories/ZArith/BinInt.v
@@ -1775,8 +1775,6 @@ weak_Zmult_plus_distr_r (now Z.mul_add_distr_pos)
 
 Definition Zne (x y:Z) := x <> y. (* TODO : to remove someday ? *)
 
-Register Zne as plugins.omega.Zne.
-
 Ltac elim_compare com1 com2 :=
   case (Dcompare (com1 ?= com2)%Z);
     [ idtac | let x := fresh "H" in
diff --git a/theories/ZArith/ZArith.v b/theories/ZArith/ZArith.v
index ed47539e1e..5490044ac7 100644
--- a/theories/ZArith/ZArith.v
+++ b/theories/ZArith/ZArith.v
@@ -16,9 +16,11 @@ Require Export ZArith_base.
 
 Require Export Zpow_def.
 
-(** Extra modules using [Omega] or [Ring]. *)
+(** Extra modules using [Ring]. *)
 
-Require Export Omega.
+Require Export OmegaLemmas.
+Require Export PreOmega.
+Require Export ZArith_hints.
 Require Export Zcomplements.
 Require Export Zpower.
 Require Export Zdiv.
diff --git a/theories/ZArith/Znat.v b/theories/ZArith/Znat.v
index 7f72d42d1f..b7cd849323 100644
--- a/theories/ZArith/Znat.v
+++ b/theories/ZArith/Znat.v
@@ -959,13 +959,6 @@ Definition inj_lt n m := proj1 (Nat2Z.inj_lt n m).
 Definition inj_ge n m := proj1 (Nat2Z.inj_ge n m).
 Definition inj_gt n m := proj1 (Nat2Z.inj_gt n m).
 
-Register neq as plugins.omega.neq.
-Register inj_eq as plugins.omega.inj_eq.
-Register inj_neq as plugins.omega.inj_neq.
-Register inj_le as plugins.omega.inj_le.
-Register inj_lt as plugins.omega.inj_lt.
-Register inj_ge as plugins.omega.inj_ge.
-Register inj_gt as plugins.omega.inj_gt.
 
 (** For the others, a Notation is fine *)
 
@@ -1033,5 +1026,3 @@ Theorem inj_minus2 : forall n m:nat, (m > n)%nat -> Z.of_nat (n - m) = 0.
 Proof.
  intros. rewrite not_le_minus_0; auto with arith.
 Qed.
-
-Register inj_minus2 as plugins.omega.inj_minus2.
diff --git a/theories/ZArith/Zorder.v b/theories/ZArith/Zorder.v
index 4c533ac458..bef9cede12 100644
--- a/theories/ZArith/Zorder.v
+++ b/theories/ZArith/Zorder.v
@@ -64,11 +64,6 @@ Proof.
   apply Z.lt_gt_cases.
 Qed.
 
-Register dec_Zne as plugins.omega.dec_Zne.
-Register dec_Zgt as plugins.omega.dec_Zgt.
-Register dec_Zge as plugins.omega.dec_Zge.
-Register not_Zeq as plugins.omega.not_Zeq.
-
 (** * Relating strict and large orders *)
 
 Notation Zgt_iff_lt := Z.gt_lt_iff (only parsing).
@@ -120,12 +115,6 @@ Proof.
   destruct (Z.eq_decidable n m); [assumption|now elim H].
 Qed.
 
-Register Znot_le_gt as plugins.omega.Znot_le_gt.
-Register Znot_lt_ge as plugins.omega.Znot_lt_ge.
-Register Znot_ge_lt as plugins.omega.Znot_ge_lt.
-Register Znot_gt_le as plugins.omega.Znot_gt_le.
-Register not_Zne as plugins.omega.not_Zne.
-
 (** * Equivalence and order properties *)
 
 (** Reflexivity *)
diff --git a/theories/ZArith/auxiliary.v b/theories/ZArith/auxiliary.v
index 10ea6cc03e..369a0c46ae 100644
--- a/theories/ZArith/auxiliary.v
+++ b/theories/ZArith/auxiliary.v
@@ -93,11 +93,3 @@ Proof.
  apply Z.le_lt_trans with (m*n+p); trivial.
  now apply Z.add_lt_mono_l.
 Qed.
-
-Register Zegal_left as plugins.omega.Zegal_left.
-Register Zne_left as plugins.omega.Zne_left.
-Register Zlt_left as plugins.omega.Zlt_left.
-Register Zgt_left as plugins.omega.Zgt_left.
-Register Zle_left as plugins.omega.Zle_left.
-Register Zge_left as plugins.omega.Zge_left.
-Register Zmult_le_approx as plugins.omega.Zmult_le_approx.
diff --git a/theories/dune b/theories/dune
index 1cd3d8c119..ced818c3ba 100644
--- a/theories/dune
+++ b/theories/dune
@@ -22,7 +22,6 @@
 
    coq-core.plugins.ring
    coq-core.plugins.nsatz
-   coq-core.plugins.omega
 
    coq-core.plugins.zify
    coq-core.plugins.micromega
diff --git a/theories/omega/Omega.v b/theories/omega/Omega.v
deleted file mode 100644
index 5c52284621..0000000000
--- a/theories/omega/Omega.v
+++ /dev/null
@@ -1,24 +0,0 @@
-(************************************************************************)
-(*         *   The Coq Proof Assistant / The Coq Development Team       *)
-(*  v      *         Copyright INRIA, CNRS and contributors             *)
-(* <O___,, * (see version control and CREDITS file for authors & dates) *)
-(*   \VV/  **************************************************************)
-(*    //   *    This file is distributed under the terms of the         *)
-(*         *     GNU Lesser General Public License Version 2.1          *)
-(*         *     (see LICENSE file for the text of the license)         *)
-(************************************************************************)
-(**************************************************************************)
-(*                                                                        *)
-(* Omega: a solver of quantifier-free problems in Presburger Arithmetic   *)
-(*                                                                        *)
-(* Pierre Crégut (CNET, Lannion, France)                                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* We import what is necessary for Omega *)
-Require Export ZArith_base.
-Require Export OmegaLemmas.
-Require Export PreOmega.
-Require Export ZArith_hints.
-
-Declare ML Module "omega_plugin".
diff --git a/theories/omega/OmegaLemmas.v b/theories/omega/OmegaLemmas.v
index 08e9ac345d..8ccd8d76f6 100644
--- a/theories/omega/OmegaLemmas.v
+++ b/theories/omega/OmegaLemmas.v
@@ -261,47 +261,3 @@ Proof.
   intros n; exists (Z.of_nat n); split; trivial.
   rewrite Z.mul_1_r, Z.add_0_r. apply Nat2Z.is_nonneg.
 Qed.
-
-Register fast_Zplus_assoc_reverse as plugins.omega.fast_Zplus_assoc_reverse.
-Register fast_Zplus_assoc as plugins.omega.fast_Zplus_assoc.
-Register fast_Zmult_assoc_reverse as plugins.omega.fast_Zmult_assoc_reverse.
-Register fast_Zplus_permute as plugins.omega.fast_Zplus_permute.
-Register fast_Zplus_comm as plugins.omega.fast_Zplus_comm.
-Register fast_Zmult_comm as plugins.omega.fast_Zmult_comm.
-
-Register OMEGA1 as plugins.omega.OMEGA1.
-Register OMEGA2 as plugins.omega.OMEGA2.
-Register OMEGA3 as plugins.omega.OMEGA3.
-Register OMEGA4 as plugins.omega.OMEGA4.
-Register OMEGA5 as plugins.omega.OMEGA5.
-Register OMEGA6 as plugins.omega.OMEGA6.
-Register OMEGA7 as plugins.omega.OMEGA7.
-Register OMEGA8 as plugins.omega.OMEGA8.
-Register OMEGA9 as plugins.omega.OMEGA9.
-Register fast_OMEGA10 as plugins.omega.fast_OMEGA10.
-Register fast_OMEGA11 as plugins.omega.fast_OMEGA11.
-Register fast_OMEGA12 as plugins.omega.fast_OMEGA12.
-Register fast_OMEGA13 as plugins.omega.fast_OMEGA13.
-Register fast_OMEGA14 as plugins.omega.fast_OMEGA14.
-Register fast_OMEGA15 as plugins.omega.fast_OMEGA15.
-Register fast_OMEGA16 as plugins.omega.fast_OMEGA16.
-Register OMEGA17 as plugins.omega.OMEGA17.
-Register OMEGA18 as plugins.omega.OMEGA18.
-Register OMEGA19 as plugins.omega.OMEGA19.
-Register OMEGA20 as plugins.omega.OMEGA20.
-
-Register fast_Zred_factor0 as plugins.omega.fast_Zred_factor0.
-Register fast_Zred_factor1 as plugins.omega.fast_Zred_factor1.
-Register fast_Zred_factor2 as plugins.omega.fast_Zred_factor2.
-Register fast_Zred_factor3 as plugins.omega.fast_Zred_factor3.
-Register fast_Zred_factor4 as plugins.omega.fast_Zred_factor4.
-Register fast_Zred_factor5 as plugins.omega.fast_Zred_factor5.
-Register fast_Zred_factor6 as plugins.omega.fast_Zred_factor6.
-
-Register fast_Zmult_plus_distr_l as plugins.omega.fast_Zmult_plus_distr_l.
-Register fast_Zopp_plus_distr as plugins.omega.fast_Zopp_plus_distr.
-Register fast_Zopp_mult_distr_r as plugins.omega.fast_Zopp_mult_distr_r.
-Register fast_Zopp_eq_mult_neg_1 as plugins.omega.fast_Zopp_eq_mult_neg_1.
-
-Register new_var as plugins.omega.new_var.
-Register intro_Z as plugins.omega.intro_Z.
diff --git a/theories/omega/OmegaPlugin.v b/theories/omega/OmegaPlugin.v
deleted file mode 100644
index e0cf24f6aa..0000000000
--- a/theories/omega/OmegaPlugin.v
+++ /dev/null
@@ -1,17 +0,0 @@
-(************************************************************************)
-(*         *   The Coq Proof Assistant / The Coq Development Team       *)
-(*  v      *         Copyright INRIA, CNRS and contributors             *)
-(* <O___,, * (see version control and CREDITS file for authors & dates) *)
-(*   \VV/  **************************************************************)
-(*    //   *    This file is distributed under the terms of the         *)
-(*         *     GNU Lesser General Public License Version 2.1          *)
-(*         *     (see LICENSE file for the text of the license)         *)
-(************************************************************************)
-
-(* To strictly import the omega tactic *)
-
-Require ZArith_base.
-Require OmegaLemmas.
-Require PreOmega.
-
-Declare ML Module "omega_plugin".
diff --git a/theories/omega/OmegaTactic.v b/theories/omega/OmegaTactic.v
deleted file mode 100644
index e0cf24f6aa..0000000000
--- a/theories/omega/OmegaTactic.v
+++ /dev/null
@@ -1,17 +0,0 @@
-(************************************************************************)
-(*         *   The Coq Proof Assistant / The Coq Development Team       *)
-(*  v      *         Copyright INRIA, CNRS and contributors             *)
-(* <O___,, * (see version control and CREDITS file for authors & dates) *)
-(*   \VV/  **************************************************************)
-(*    //   *    This file is distributed under the terms of the         *)
-(*         *     GNU Lesser General Public License Version 2.1          *)
-(*         *     (see LICENSE file for the text of the license)         *)
-(************************************************************************)
-
-(* To strictly import the omega tactic *)
-
-Require ZArith_base.
-Require OmegaLemmas.
-Require PreOmega.
-
-Declare ML Module "omega_plugin".
diff --git a/theories/omega/PreOmega.v b/theories/omega/PreOmega.v
index 70f25e7243..bf873785d0 100644
--- a/theories/omega/PreOmega.v
+++ b/theories/omega/PreOmega.v
@@ -12,9 +12,10 @@ Require Import Arith Max Min BinInt BinNat Znat Nnat.
 
 Local Open Scope Z_scope.
 
-(** * [Z.div_mod_to_equations], [Z.quot_rem_to_equations], [Z.to_euclidean_division_equations]: the tactics for preprocessing [Z.div] and [Z.modulo], [Z.quot] and [Z.rem] *)
+(** * [Z.div_mod_to_equations], [Z.quot_rem_to_equations], [Z.to_euclidean_division_equations]:
+     the tactics for preprocessing [Z.div] and [Z.modulo], [Z.quot] and [Z.rem] *)
 
-(** These tactic use the complete specification of [Z.div] and
+(** These tactics use the complete specification of [Z.div] and
     [Z.modulo] ([Z.quot] and [Z.rem], respectively) to remove these
     functions from the goal without losing information.  The
     [Z.euclidean_division_equations_cleanup] tactic removes needless
@@ -127,449 +128,6 @@ Module Z.
   Ltac to_euclidean_division_equations := div_mod_to_equations'; quot_rem_to_equations'; euclidean_division_equations_cleanup.
 End Z.
 
-Set Warnings "-deprecated-tactic".
-
-(** * zify: the Z-ification tactic *)
-
-(* This tactic searches for nat and N and positive elements in the goal and
-   translates everything into Z. It is meant as a pre-processor for
-   (r)omega; for instance a positivity hypothesis is added whenever
-     - a multiplication is encountered
-     - an atom is encountered (that is a variable or an unknown construct)
-
-   Recognized relations (can be handled as deeply as allowed by setoid rewrite):
-     - { eq, le, lt, ge, gt } on { Z, positive, N, nat }
-
-   Recognized operations:
-     - on Z: Z.min, Z.max, Z.abs, Z.sgn are translated in term of <= < =
-     - on nat: + * - S O pred min max Pos.to_nat N.to_nat Z.abs_nat
-     - on positive: Zneg Zpos xI xO xH + * - Pos.succ Pos.pred Pos.min Pos.max Pos.of_succ_nat
-     - on N: N0 Npos + * - N.pred N.succ N.min N.max N.of_nat Z.abs_N
-*)
-
-
-
-
-(** I) translation of Z.max, Z.min, Z.abs, Z.sgn into recognized equations *)
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_unop_core t thm a :=
- (* Let's introduce the specification theorem for t *)
- pose proof (thm a);
- (* Then we replace (t a) everywhere with a fresh variable *)
- let z := fresh "z" in set (z:=t a) in *; clearbody z.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_unop_var_or_term t thm a :=
- (* If a is a variable, no need for aliasing *)
- let za := fresh "z" in
- (rename a into za; rename za into a; zify_unop_core t thm a) ||
- (* Otherwise, a is a complex term: we alias it. *)
- (remember a as za; zify_unop_core t thm za).
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_unop t thm a :=
- (* If a is a scalar, we can simply reduce the unop. *)
- (* Note that simpl wasn't enough to reduce [Z.max 0 0] (#5439) *)
- let isz := isZcst a in
- match isz with
-  | true =>
-    let u := eval compute in (t a) in
-    change (t a) with u in *
-  | _ => zify_unop_var_or_term t thm a
- end.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_unop_nored t thm a :=
- (* in this version, we don't try to reduce the unop (that can be (Z.add x)) *)
- let isz := isZcst a in
- match isz with
-  | true => zify_unop_core t thm a
-  | _ => zify_unop_var_or_term t thm a
- end.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_binop t thm a b:=
- (* works as zify_unop, except that we should be careful when
-    dealing with b, since it can be equal to a *)
- let isza := isZcst a in
- match isza with
-   | true => zify_unop (t a) (thm a) b
-   | _ =>
-       let za := fresh "z" in
-       (rename a into za; rename za into a; zify_unop_nored (t a) (thm a) b) ||
-       (remember a as za; match goal with
-         | H : za = b |- _ => zify_unop_nored (t za) (thm za) za
-         | _ => zify_unop_nored (t za) (thm za) b
-        end)
- end.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_op_1 :=
-  match goal with
-   | x := ?t : Z |- _ => let h := fresh "heq_" x in pose proof (eq_refl : x = t) as h; clearbody x
-   | |- context [ Z.max ?a ?b ] => zify_binop Z.max Z.max_spec a b
-   | H : context [ Z.max ?a ?b ] |- _ => zify_binop Z.max Z.max_spec a b
-   | |- context [ Z.min ?a ?b ] => zify_binop Z.min Z.min_spec a b
-   | H : context [ Z.min ?a ?b ] |- _ => zify_binop Z.min Z.min_spec a b
-   | |- context [ Z.sgn ?a ] => zify_unop Z.sgn Z.sgn_spec a
-   | H : context [ Z.sgn ?a ] |- _ => zify_unop Z.sgn Z.sgn_spec a
-   | |- context [ Z.abs ?a ] => zify_unop Z.abs Z.abs_spec a
-   | H : context [ Z.abs ?a ] |- _ => zify_unop Z.abs Z.abs_spec a
-  end.
-
-Ltac zify_op := repeat zify_op_1.
-
-
-(** II) Conversion from nat to Z *)
-
-
-Definition Z_of_nat' := Z.of_nat.
-
-Ltac hide_Z_of_nat t :=
-  let z := fresh "z" in set (z:=Z.of_nat t) in *;
-  change Z.of_nat with Z_of_nat' in z;
-  unfold z in *; clear z.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_nat_rel :=
- match goal with
-  (* I: equalities *)
-  | x := ?t : nat |- _ => let h := fresh "heq_" x in pose proof (eq_refl : x = t) as h; clearbody x
-  | |- (@eq nat ?a ?b) => apply (Nat2Z.inj a b) (* shortcut *)
-  | H : context [ @eq nat ?a ?b ] |- _ => rewrite <- (Nat2Z.inj_iff a b) in H
-  | |- context [ @eq nat ?a ?b ] =>       rewrite <- (Nat2Z.inj_iff a b)
-  (* II: less than *)
-  | H : context [ lt ?a ?b ] |- _ => rewrite (Nat2Z.inj_lt a b) in H
-  | |- context [ lt ?a ?b ] =>       rewrite (Nat2Z.inj_lt a b)
-  (* III: less or equal *)
-  | H : context [ le ?a ?b ] |- _ => rewrite (Nat2Z.inj_le a b) in H
-  | |- context [ le ?a ?b ] =>       rewrite (Nat2Z.inj_le a b)
-  (* IV: greater than *)
-  | H : context [ gt ?a ?b ] |- _ => rewrite (Nat2Z.inj_gt a b) in H
-  | |- context [ gt ?a ?b ] =>       rewrite (Nat2Z.inj_gt a b)
-  (* V: greater or equal *)
-  | H : context [ ge ?a ?b ] |- _ => rewrite (Nat2Z.inj_ge a b) in H
-  | |- context [ ge ?a ?b ] =>       rewrite (Nat2Z.inj_ge a b)
- end.
-
-Ltac zify_nat_op :=
- match goal with
-  (* misc type conversions: positive/N/Z to nat *)
-  | H : context [ Z.of_nat (Pos.to_nat ?a) ] |- _ => rewrite (positive_nat_Z a) in H
-  | |- context [ Z.of_nat (Pos.to_nat ?a) ] => rewrite (positive_nat_Z a)
-  | H : context [ Z.of_nat (N.to_nat ?a) ] |- _ => rewrite (N_nat_Z a) in H
-  | |- context [ Z.of_nat (N.to_nat ?a) ] => rewrite (N_nat_Z a)
-  | H : context [ Z.of_nat (Z.abs_nat ?a) ] |- _ => rewrite (Zabs2Nat.id_abs a) in H
-  | |- context [ Z.of_nat (Z.abs_nat ?a) ] => rewrite (Zabs2Nat.id_abs a)
-
-  (* plus -> Z.add *)
-  | H : context [ Z.of_nat (plus ?a ?b) ] |- _ => rewrite (Nat2Z.inj_add a b) in H
-  | |- context [ Z.of_nat (plus ?a ?b) ] => rewrite (Nat2Z.inj_add a b)
-
-  (* min -> Z.min *)
-  | H : context [ Z.of_nat (min ?a ?b) ] |- _ => rewrite (Nat2Z.inj_min a b) in H
-  | |- context [ Z.of_nat (min ?a ?b) ] => rewrite (Nat2Z.inj_min a b)
-
-  (* max -> Z.max *)
-  | H : context [ Z.of_nat (max ?a ?b) ] |- _ => rewrite (Nat2Z.inj_max a b) in H
-  | |- context [ Z.of_nat (max ?a ?b) ] => rewrite (Nat2Z.inj_max a b)
-
-  (* minus -> Z.max (Z.sub ... ...) 0 *)
-  | H : context [ Z.of_nat (minus ?a ?b) ] |- _ => rewrite (Nat2Z.inj_sub_max a b) in H
-  | |- context [ Z.of_nat (minus ?a ?b) ] => rewrite (Nat2Z.inj_sub_max a b)
-
-  (* pred -> minus ... -1 -> Z.max (Z.sub ... -1) 0 *)
-  | H : context [ Z.of_nat (pred ?a) ] |- _ => rewrite (pred_of_minus a) in H
-  | |- context [ Z.of_nat (pred ?a) ] => rewrite (pred_of_minus a)
-
-  (* mult -> Z.mul and a positivity hypothesis *)
-  | H : context [ Z.of_nat (mult ?a ?b) ] |- _ =>
-        pose proof (Nat2Z.is_nonneg (mult a b));
-        rewrite (Nat2Z.inj_mul a b) in *
-  | |- context [ Z.of_nat (mult ?a ?b) ] =>
-        pose proof (Nat2Z.is_nonneg (mult a b));
-        rewrite (Nat2Z.inj_mul a b) in *
-
-  (* O -> Z0 *)
-  | H : context [ Z.of_nat O ] |- _ => change (Z.of_nat O) with Z0 in H
-  | |- context [ Z.of_nat O ] => change (Z.of_nat O) with Z0
-
-  (* S -> number or Z.succ *)
-  | H : context [ Z.of_nat (S ?a) ] |- _ =>
-     let isnat := isnatcst a in
-     match isnat with
-      | true =>
-        let t := eval compute in (Z.of_nat (S a)) in
-        change (Z.of_nat (S a)) with t in H
-      | _ => rewrite (Nat2Z.inj_succ a) in H
-      | _ => (* if the [rewrite] fails (most likely a dependent occurrence of [Z.of_nat (S a)]),
-                hide [Z.of_nat (S a)] in this one hypothesis *)
-        change (Z.of_nat (S a)) with (Z_of_nat' (S a)) in H
-     end
-  | |- context [ Z.of_nat (S ?a) ] =>
-     let isnat := isnatcst a in
-     match isnat with
-      | true =>
-        let t := eval compute in (Z.of_nat (S a)) in
-        change (Z.of_nat (S a)) with t
-      | _ => rewrite (Nat2Z.inj_succ a)
-      | _ => (* if the [rewrite] fails (most likely a dependent occurrence of [Z.of_nat (S a)]),
-                hide [Z.of_nat (S a)] in the goal *)
-        change (Z.of_nat (S a)) with (Z_of_nat' (S a))
-     end
-
-  (* atoms of type nat : we add a positivity condition (if not already there) *)
-  | _ : 0 <= Z.of_nat ?a |- _ => hide_Z_of_nat a
-  | _ : context [ Z.of_nat ?a ] |- _ =>
-    pose proof (Nat2Z.is_nonneg a); hide_Z_of_nat a
-  | |- context [ Z.of_nat ?a ] =>
-    pose proof (Nat2Z.is_nonneg a); hide_Z_of_nat a
- end.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_nat := repeat zify_nat_rel; repeat zify_nat_op; unfold Z_of_nat' in *.
-
-(* III) conversion from positive to Z *)
-
-Definition Zpos' := Zpos.
-Definition Zneg' := Zneg.
-
-Ltac hide_Zpos t :=
-  let z := fresh "z" in set (z:=Zpos t) in *;
-  change Zpos with Zpos' in z;
-  unfold z in *; clear z.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_positive_rel :=
- match goal with
-  (* I: equalities *)
-  | x := ?t : positive |- _ => let h := fresh "heq_" x in pose proof (eq_refl : x = t) as h; clearbody x
-  | |- (@eq positive ?a ?b) => apply Pos2Z.inj
-  | H : context [ @eq positive ?a ?b ] |- _ => rewrite <- (Pos2Z.inj_iff a b) in H
-  | |- context [ @eq positive ?a ?b ] =>       rewrite <- (Pos2Z.inj_iff a b)
-  (* II: less than *)
-  | H : context [ (?a < ?b)%positive ] |- _ => change (a<b)%positive with (Zpos a<Zpos b) in H
-  | |- context [ (?a < ?b)%positive ] => change (a<b)%positive with (Zpos a<Zpos b)
-  (* III: less or equal *)
-  | H : context [ (?a <= ?b)%positive ] |- _ => change (a<=b)%positive with (Zpos a<=Zpos b) in H
-  | |- context [ (?a <= ?b)%positive ] => change (a<=b)%positive with (Zpos a<=Zpos b)
-  (* IV: greater than *)
-  | H : context [ (?a > ?b)%positive ] |- _ => change (a>b)%positive with (Zpos a>Zpos b) in H
-  | |- context [ (?a > ?b)%positive ] => change (a>b)%positive with (Zpos a>Zpos b)
-  (* V: greater or equal *)
-  | H : context [ (?a >= ?b)%positive ] |- _ => change (a>=b)%positive with (Zpos a>=Zpos b) in H
-  | |- context [ (?a >= ?b)%positive ] => change (a>=b)%positive with (Zpos a>=Zpos b)
- end.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_positive_op :=
- match goal with
- (* Z.pow_pos -> Z.pow *)
- | H : context [ Z.pow_pos ?a ?b ] |- _ => change (Z.pow_pos a b) with (Z.pow a (Z.pos b)) in H
- | |- context [ Z.pow_pos ?a ?b ]  => change (Z.pow_pos a b) with (Z.pow a (Z.pos b))
- (* Zneg -> -Zpos (except for numbers) *)
-  | H : context [ Zneg ?a ] |- _ =>
-     let isp := isPcst a in
-     match isp with
-      | true => change (Zneg a) with (Zneg' a) in H
-      | _ => change (Zneg a) with (- Zpos a) in H
-     end
-  | |- context [ Zneg ?a ] =>
-     let isp := isPcst a in
-     match isp with
-      | true => change (Zneg a) with (Zneg' a)
-      | _ => change (Zneg a) with (- Zpos a)
-     end
-
-  (* misc type conversions: nat to positive *)
-  | H : context [ Zpos (Pos.of_succ_nat ?a) ] |- _ => rewrite (Zpos_P_of_succ_nat a) in H
-  | |- context [ Zpos (Pos.of_succ_nat ?a) ] => rewrite (Zpos_P_of_succ_nat a)
-
-  (* Z.power_pos *)
-  | H : context [ Zpos (Pos.of_succ_nat ?a) ] |- _ => rewrite (Zpos_P_of_succ_nat a) in H
-  | |- context [ Zpos (Pos.of_succ_nat ?a) ] => rewrite (Zpos_P_of_succ_nat a)
-
-  (* Pos.add -> Z.add *)
-  | H : context [ Zpos (?a + ?b) ] |- _ => change (Zpos (a+b)) with (Zpos a + Zpos b) in H
-  | |- context [ Zpos (?a + ?b) ] => change (Zpos (a+b)) with (Zpos a + Zpos b)
-
-  (* Pos.min -> Z.min *)
-  | H : context [ Zpos (Pos.min ?a ?b) ] |- _ => rewrite (Pos2Z.inj_min a b) in H
-  | |- context [ Zpos (Pos.min ?a ?b) ] => rewrite (Pos2Z.inj_min a b)
-
-  (* Pos.max -> Z.max *)
-  | H : context [ Zpos (Pos.max ?a ?b) ] |- _ => rewrite (Pos2Z.inj_max a b) in H
-  | |- context [ Zpos (Pos.max ?a ?b) ] => rewrite (Pos2Z.inj_max a b)
-
-  (* Pos.sub -> Z.max 1 (Z.sub ... ...) *)
-  | H : context [ Zpos (Pos.sub ?a ?b) ] |- _ => rewrite (Pos2Z.inj_sub_max a b) in H
-  | |- context [ Zpos (Pos.sub ?a ?b) ] => rewrite (Pos2Z.inj_sub_max a b)
-
-  (* Pos.succ -> Z.succ *)
-  | H : context [ Zpos (Pos.succ ?a) ] |- _ => rewrite (Pos2Z.inj_succ a) in H
-  | |- context [ Zpos (Pos.succ ?a) ] => rewrite (Pos2Z.inj_succ a)
-
-  (* Pos.pred -> Pos.sub ... -1 -> Z.max 1 (Z.sub ... - 1) *)
-  | H : context [ Zpos (Pos.pred ?a) ] |- _ => rewrite <- (Pos.sub_1_r a) in H
-  | |- context [ Zpos (Pos.pred ?a) ] => rewrite <- (Pos.sub_1_r a)
-
-  (* Pos.mul -> Z.mul and a positivity hypothesis *)
-  | H : context [ Zpos (?a * ?b) ] |- _ =>
-        pose proof (Pos2Z.is_pos (Pos.mul a b));
-        change (Zpos (a*b)) with (Zpos a * Zpos b) in *
-  | |- context [ Zpos (?a * ?b) ] =>
-        pose proof (Pos2Z.is_pos (Pos.mul a b));
-        change (Zpos (a*b)) with (Zpos a * Zpos b) in *
-
-  (* xO *)
-  | H : context [ Zpos (xO ?a) ] |- _ =>
-     let isp := isPcst a in
-     match isp with
-      | true => change (Zpos (xO a)) with (Zpos' (xO a)) in H
-      | _ => rewrite (Pos2Z.inj_xO a) in H
-     end
-  | |- context [ Zpos (xO ?a) ] =>
-     let isp := isPcst a in
-     match isp with
-      | true => change (Zpos (xO a)) with (Zpos' (xO a))
-      | _ => rewrite (Pos2Z.inj_xO a)
-     end
-  (* xI *)
-  | H : context [ Zpos (xI ?a) ] |- _ =>
-     let isp := isPcst a in
-     match isp with
-      | true => change (Zpos (xI a)) with (Zpos' (xI a)) in H
-      | _ => rewrite (Pos2Z.inj_xI a) in H
-     end
-  | |- context [ Zpos (xI ?a) ] =>
-     let isp := isPcst a in
-     match isp with
-      | true => change (Zpos (xI a)) with (Zpos' (xI a))
-      | _ => rewrite (Pos2Z.inj_xI a)
-     end
-
-  (* xI : nothing to do, just prevent adding a useless positivity condition *)
-  | H : context [ Zpos xH ] |- _ => hide_Zpos xH
-  | |- context [ Zpos xH ] => hide_Zpos xH
-
-  (* atoms of type positive : we add a positivity condition (if not already there) *)
-  | _ : 0 < Zpos ?a |- _ => hide_Zpos a
-  | _ : context [ Zpos ?a ] |- _ => pose proof (Pos2Z.is_pos a); hide_Zpos a
-  | |- context [ Zpos ?a ] => pose proof (Pos2Z.is_pos a); hide_Zpos a
- end.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_positive :=
- repeat zify_positive_rel; repeat zify_positive_op; unfold Zpos',Zneg' in *.
-
-
-
-
-
-(* IV) conversion from N to Z *)
-
-Definition Z_of_N' := Z.of_N.
-
-Ltac hide_Z_of_N t :=
-  let z := fresh "z" in set (z:=Z.of_N t) in *;
-  change Z.of_N with Z_of_N' in z;
-  unfold z in *; clear z.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_N_rel :=
- match goal with
-  (* I: equalities *)
-  | x := ?t : N |- _ => let h := fresh "heq_" x in pose proof (eq_refl : x = t) as h; clearbody x
-  | |- (@eq N ?a ?b) => apply (N2Z.inj a b) (* shortcut *)
-  | H : context [ @eq N ?a ?b ] |- _ => rewrite <- (N2Z.inj_iff a b) in H
-  | |- context [ @eq N ?a ?b ] =>       rewrite <- (N2Z.inj_iff a b)
-  (* II: less than *)
-  | H : context [ (?a < ?b)%N ] |- _ => rewrite (N2Z.inj_lt a b) in H
-  | |- context [ (?a < ?b)%N ] =>       rewrite (N2Z.inj_lt a b)
-  (* III: less or equal *)
-  | H : context [ (?a <= ?b)%N ] |- _ => rewrite (N2Z.inj_le a b) in H
-  | |- context [ (?a <= ?b)%N ] =>       rewrite (N2Z.inj_le a b)
-  (* IV: greater than *)
-  | H : context [ (?a > ?b)%N ] |- _ => rewrite (N2Z.inj_gt a b) in H
-  | |- context [ (?a > ?b)%N ] =>       rewrite (N2Z.inj_gt a b)
-  (* V: greater or equal *)
-  | H : context [ (?a >= ?b)%N ] |- _ => rewrite (N2Z.inj_ge a b) in H
-  | |- context [ (?a >= ?b)%N ] =>       rewrite (N2Z.inj_ge a b)
- end.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_N_op :=
- match goal with
-  (* misc type conversions: nat to positive *)
-  | H : context [ Z.of_N (N.of_nat ?a) ] |- _ => rewrite (nat_N_Z a) in H
-  | |- context [ Z.of_N (N.of_nat ?a) ] => rewrite (nat_N_Z a)
-  | H : context [ Z.of_N (Z.abs_N ?a) ] |- _ => rewrite (N2Z.inj_abs_N a) in H
-  | |- context [ Z.of_N (Z.abs_N ?a) ] => rewrite (N2Z.inj_abs_N a)
-  | H : context [ Z.of_N (Npos ?a) ] |- _ => rewrite (N2Z.inj_pos a) in H
-  | |- context [ Z.of_N (Npos ?a) ] => rewrite (N2Z.inj_pos a)
-  | H : context [ Z.of_N N0 ] |- _ => change (Z.of_N N0) with Z0 in H
-  | |- context [ Z.of_N N0 ] => change (Z.of_N N0) with Z0
-
-  (* N.add -> Z.add *)
-  | H : context [ Z.of_N (N.add ?a ?b) ] |- _ => rewrite (N2Z.inj_add a b) in H
-  | |- context [ Z.of_N (N.add ?a ?b) ] => rewrite (N2Z.inj_add a b)
-
-  (* N.min -> Z.min *)
-  | H : context [ Z.of_N (N.min ?a ?b) ] |- _ => rewrite (N2Z.inj_min a b) in H
-  | |- context [ Z.of_N (N.min ?a ?b) ] => rewrite (N2Z.inj_min a b)
-
-  (* N.max -> Z.max *)
-  | H : context [ Z.of_N (N.max ?a ?b) ] |- _ => rewrite (N2Z.inj_max a b) in H
-  | |- context [ Z.of_N (N.max ?a ?b) ] => rewrite (N2Z.inj_max a b)
-
-  (* N.sub -> Z.max 0 (Z.sub ... ...) *)
-  | H : context [ Z.of_N (N.sub ?a ?b) ] |- _ => rewrite (N2Z.inj_sub_max a b) in H
-  | |- context [ Z.of_N (N.sub ?a ?b) ] => rewrite (N2Z.inj_sub_max a b)
-
-  (* pred -> minus ... -1 -> Z.max (Z.sub ... -1) 0 *)
-  | H : context [ Z.of_N (N.pred ?a) ] |- _ => rewrite (N.pred_sub a) in H
-  | |- context [ Z.of_N (N.pred ?a) ] => rewrite (N.pred_sub a)
-
-  (* N.succ -> Z.succ *)
-  | H : context [ Z.of_N (N.succ ?a) ] |- _ => rewrite (N2Z.inj_succ a) in H
-  | |- context [ Z.of_N (N.succ ?a) ] => rewrite (N2Z.inj_succ a)
-
-  (* N.mul -> Z.mul and a positivity hypothesis *)
-  | H : context [ Z.of_N (N.mul ?a ?b) ] |- _ =>
-        pose proof (N2Z.is_nonneg (N.mul a b)); rewrite (N2Z.inj_mul a b) in *
-  | |- context [ Z.of_N  (N.mul ?a ?b) ] =>
-        pose proof (N2Z.is_nonneg (N.mul a b)); rewrite (N2Z.inj_mul a b) in *
-
-  (* N.div -> Z.div and a positivity hypothesis *)
-  | H : context [ Z.of_N (N.div ?a ?b) ] |- _ =>
-        pose proof (N2Z.is_nonneg (N.div a b)); rewrite (N2Z.inj_div a b) in *
-  | |- context [ Z.of_N  (N.div ?a ?b) ] =>
-        pose proof (N2Z.is_nonneg (N.div a b)); rewrite (N2Z.inj_div a b) in *
-
-  (* N.modulo -> Z.rem / Z.modulo and a positivity hypothesis (N.modulo agrees with Z.modulo on everything except 0; so we pose both the non-zero proof for this agreement, but also replace things with [Z.rem]) *)
-  | H : context [ Z.of_N (N.modulo ?a ?b) ] |- _ =>
-    pose proof (N2Z.is_nonneg (N.modulo a b));
-    pose proof (@Z.quot_div_nonneg (Z.of_N a) (Z.of_N b) (N2Z.is_nonneg a));
-    pose proof (@Z.rem_mod_nonneg (Z.of_N a) (Z.of_N b) (N2Z.is_nonneg a));
-    rewrite (N2Z.inj_rem a b) in *
-  | |- context [ Z.of_N  (N.div ?a ?b) ] =>
-    pose proof (N2Z.is_nonneg (N.modulo a b));
-    pose proof (@Z.quot_div_nonneg (Z.of_N a) (Z.of_N b) (N2Z.is_nonneg a));
-    pose proof (@Z.rem_mod_nonneg (Z.of_N a) (Z.of_N b) (N2Z.is_nonneg a));
-    rewrite (N2Z.inj_rem a b) in *
-
-  (* atoms of type N : we add a positivity condition (if not already there) *)
-  | _ : 0 <= Z.of_N ?a |- _ => hide_Z_of_N a
-  | _ : context [ Z.of_N ?a ] |- _ => pose proof (N2Z.is_nonneg a); hide_Z_of_N a
-  | |- context [ Z.of_N ?a ] => pose proof (N2Z.is_nonneg a); hide_Z_of_N a
- end.
-
-#[deprecated( note = "Use 'zify' instead")]
-Ltac zify_N := repeat zify_N_rel; repeat zify_N_op; unfold Z_of_N' in *.
-
-(** The complete Z-ification tactic *)
-
 Require  Import ZifyClasses ZifyInst.
 Require  Zify.