Imported Upstream version 8.0pl1upstream/8.0pl1

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diff --git a/contrib/romega/README b/contrib/romega/README
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+This work was done for the RNRT Project Calife. 
+As such it is distributed under the LGPL licence.
+
+Report bugs to :
+  pierre.cregut@francetelecom.com
+
diff --git a/contrib/romega/ROmega.v b/contrib/romega/ROmega.v
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+++ b/contrib/romega/ROmega.v
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+(*************************************************************************
+
+   PROJET RNRT Calife - 2001
+   Author: Pierre Crégut - France Télécom R&D
+   Licence : LGPL version 2.1
+
+ *************************************************************************)
+
+Require Import Omega.
+Require Import ReflOmegaCore.
+
diff --git a/contrib/romega/ReflOmegaCore.v b/contrib/romega/ReflOmegaCore.v
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+(*************************************************************************
+
+   PROJET RNRT Calife - 2001
+   Author: Pierre Crégut - France Télécom R&D
+   Licence du projet : LGPL version 2.1
+
+ *************************************************************************)
+
+Require Import Arith.
+Require Import List.
+Require Import Bool.
+Require Import ZArith.
+Require Import OmegaLemmas.
+
+(* \subsection{Definition of basic types} *)
+
+(* \subsubsection{Environment of propositions (lists) *)
+Inductive PropList : Type :=
+  | Pnil : PropList
+  | Pcons : Prop -> PropList -> PropList.
+
+(* Access function for the environment with a default *)
+Fixpoint nthProp (n : nat) (l : PropList) (default : Prop) {struct l} :
+ Prop :=
+  match n, l with
+  | O, Pcons x l' => x
+  | O, other => default
+  | S m, Pnil => default
+  | S m, Pcons x t => nthProp m t default
+  end.
+   
+(* \subsubsection{Définition of reified integer expressions}
+   Terms are either:
+   \begin{itemize}
+   \item integers [Tint]
+   \item variables [Tvar]
+   \item operation over integers (addition, product, opposite, subtraction)
+   The last two are translated in additions and products. *)
+
+Inductive term : Set :=
+  | Tint : Z -> term
+  | Tplus : term -> term -> term
+  | Tmult : term -> term -> term
+  | Tminus : term -> term -> term
+  | Topp : term -> term
+  | Tvar : nat -> term.
+
+(* \subsubsection{Definition of reified goals} *)
+(* Very restricted definition of handled predicates that should be extended
+   to cover a wider set of operations.
+   Taking care of negations and disequations require solving more than a
+   goal in parallel. This is a major improvement over previous versions. *)
+
+Inductive proposition : Set :=
+  | EqTerm : term -> term -> proposition (* egalité entre termes *)
+  | LeqTerm : term -> term -> proposition (* plus petit ou egal *)
+  | TrueTerm : proposition (* vrai *)
+  | FalseTerm : proposition (* faux *)
+  | Tnot : proposition -> proposition (* négation *)
+  | GeqTerm : term -> term -> proposition
+  | GtTerm : term -> term -> proposition
+  | LtTerm : term -> term -> proposition
+  | NeqTerm : term -> term -> proposition
+  | Tor : proposition -> proposition -> proposition
+  | Tand : proposition -> proposition -> proposition
+  | Timp : proposition -> proposition -> proposition
+  | Tprop : nat -> proposition.
+
+(* Definition of goals as a list of hypothesis *)
+Notation hyps := (list proposition) (only parsing).      
+
+(* Definition of lists of subgoals  (set of open goals) *)
+Notation lhyps := (list (list proposition)) (only parsing).
+
+(* a syngle goal packed in a subgoal list *)
+Notation singleton := (fun a : list proposition => a :: nil) (only parsing).
+
+(* an absurd goal *)
+Definition absurd := FalseTerm :: nil.
+
+(* \subsubsection{Traces for merging equations}
+   This inductive type describes how the monomial of two equations should be
+   merged when the equations are added.
+
+   For [F_equal], both equations have the same head variable and coefficient
+   must be added, furthermore if coefficients are opposite, [F_cancel] should
+   be used to collapse the term. [F_left] and [F_right] indicate which monomial
+   should be put first in the result *)
+
+Inductive t_fusion : Set :=
+  | F_equal : t_fusion
+  | F_cancel : t_fusion
+  | F_left : t_fusion
+  | F_right : t_fusion.
+
+(* \subsubsection{Rewriting steps to normalize terms} *)
+Inductive step : Set := 
+ (* apply the rewriting steps to both subterms of an operation *)
+  | C_DO_BOTH :
+      step -> step -> step
+              (* apply the rewriting step to the first branch *)
+  | C_LEFT : step -> step
+             (* apply the rewriting step to the second branch *)
+  | C_RIGHT : step -> step
+              (* apply two steps consecutively to a term *)
+  | C_SEQ : step -> step -> step
+                    (* empty step *)
+  | C_NOP : step
+      (* the following operations correspond to actual rewriting *)
+  | C_OPP_PLUS : step
+  | C_OPP_OPP : step
+  | C_OPP_MULT_R : step
+  | C_OPP_ONE :
+      step
+      (* This is a special step that reduces the term (computation) *)
+  | C_REDUCE : step
+  | C_MULT_PLUS_DISTR : step
+  | C_MULT_OPP_LEFT : step
+  | C_MULT_ASSOC_R : step
+  | C_PLUS_ASSOC_R : step
+  | C_PLUS_ASSOC_L : step
+  | C_PLUS_PERMUTE : step
+  | C_PLUS_SYM : step
+  | C_RED0 : step
+  | C_RED1 : step
+  | C_RED2 : step
+  | C_RED3 : step
+  | C_RED4 : step
+  | C_RED5 : step
+  | C_RED6 : step
+  | C_MULT_ASSOC_REDUCED : step
+  | C_MINUS : step
+  | C_MULT_SYM : step.
+
+(* \subsubsection{Omega steps} *)
+(* The following inductive type describes steps as they can be found in
+   the trace coming from the decision procedure Omega. *)
+
+Inductive t_omega : Set :=
+ (* n = 0 n!= 0 *)
+  | O_CONSTANT_NOT_NUL : nat -> t_omega
+  | O_CONSTANT_NEG :
+      nat -> t_omega
+      (* division et approximation of an equation *)
+  | O_DIV_APPROX :
+      Z ->
+      Z ->
+      term ->
+      nat ->
+      t_omega -> nat -> t_omega
+                 (* no solution because no exact division *)
+  | O_NOT_EXACT_DIVIDE :
+      Z -> Z -> term -> nat -> nat -> t_omega
+                               (* exact division *)
+  | O_EXACT_DIVIDE : Z -> term -> nat -> t_omega -> nat -> t_omega
+  | O_SUM : Z -> nat -> Z -> nat -> list t_fusion -> t_omega -> t_omega
+  | O_CONTRADICTION : nat -> nat -> nat -> t_omega
+  | O_MERGE_EQ : nat -> nat -> nat -> t_omega -> t_omega
+  | O_SPLIT_INEQ : nat -> nat -> t_omega -> t_omega -> t_omega
+  | O_CONSTANT_NUL : nat -> t_omega
+  | O_NEGATE_CONTRADICT : nat -> nat -> t_omega
+  | O_NEGATE_CONTRADICT_INV : nat -> nat -> nat -> t_omega
+  | O_STATE : Z -> step -> nat -> nat -> t_omega -> t_omega.
+
+(* \subsubsection{Règles pour normaliser les hypothèses} *)
+(* Ces règles indiquent comment normaliser les propositions utiles 
+   de chaque hypothèse utile avant la décomposition des hypothèses et 
+   incluent l'étape d'inversion pour la suppression des négations *)
+Inductive p_step : Set :=
+  | P_LEFT : p_step -> p_step
+  | P_RIGHT : p_step -> p_step
+  | P_INVERT : step -> p_step
+  | P_STEP : step -> p_step
+  | P_NOP : p_step.
+(* Liste des normalisations a effectuer : avec un constructeur dans le
+   type [p_step] permettant
+   de parcourir à la fois les branches gauches et droit, on pourrait n'avoir
+   qu'une normalisation par hypothèse. Et comme toutes les hypothèses sont 
+   utiles (sinon on ne les incluerait pas), on pourrait remplacer [h_step] 
+   par une simple liste *)
+
+Inductive h_step : Set :=
+    pair_step : nat -> p_step -> h_step.
+
+(* \subsubsection{Règles pour décomposer les hypothèses} *)
+(* Ce type permet de se diriger dans les constructeurs logiques formant les
+   prédicats des hypothèses pour aller les décomposer. Ils permettent
+   en particulier d'extraire une hypothèse d'une conjonction avec
+   éventuellement le bon niveau de négations. *)
+
+Inductive direction : Set :=
+  | D_left : direction
+  | D_right : direction
+  | D_mono : direction.
+
+(* Ce type permet d'extraire les composants utiles des hypothèses : que ce
+   soit des hypothèses générées par éclatement d'une disjonction, ou 
+   des équations. Le constructeur terminal indique comment résoudre le système
+   obtenu en recourrant au type de trace d'Omega [t_omega] *)
+
+Inductive e_step : Set :=
+  | E_SPLIT : nat -> list direction -> e_step -> e_step -> e_step
+  | E_EXTRACT : nat -> list direction -> e_step -> e_step
+  | E_SOLVE : t_omega -> e_step.
+
+(* \subsection{Egalité décidable efficace} *)
+(* Pour chaque type de donnée réifié, on calcule un test d'égalité efficace.
+   Ce n'est pas le cas de celui rendu par [Decide Equality].
+   
+   Puis on prouve deux théorèmes permettant d'éliminer de telles égalités :
+   \begin{verbatim}
+   (t1,t2: typ) (eq_typ t1 t2) = true -> t1 = t2.
+   (t1,t2: typ) (eq_typ t1 t2) = false -> ~ t1 = t2.
+   \end{verbatim} *)
+
+(* Ces deux tactiques permettent de résoudre pas mal de cas. L'une pour
+   les théorèmes positifs, l'autre pour les théorèmes négatifs *)
+
+Ltac absurd_case := simpl in |- *; intros; discriminate.
+Ltac trivial_case := unfold not in |- *; intros; discriminate.
+
+(* \subsubsection{Entiers naturels} *)
+
+Fixpoint eq_nat (t1 t2 : nat) {struct t2} : bool :=
+  match t1 with
+  | O => match t2 with
+         | O => true
+         | _ => false
+         end
+  | S n1 => match t2 with
+            | O => false
+            | S n2 => eq_nat n1 n2
+            end
+  end.
+
+Theorem eq_nat_true : forall t1 t2 : nat, eq_nat t1 t2 = true -> t1 = t2.
+
+simple induction t1;
+ [ intro t2; case t2; [ trivial | absurd_case ]
+ | intros n H t2; case t2;
+    [ absurd_case
+    | simpl in |- *; intros; rewrite (H n0); [ trivial | assumption ] ] ].
+
+Qed.
+  
+Theorem eq_nat_false : forall t1 t2 : nat, eq_nat t1 t2 = false -> t1 <> t2.
+
+simple induction t1;
+ [ intro t2; case t2; [ simpl in |- *; intros; discriminate | trivial_case ]
+ | intros n H t2; case t2; simpl in |- *; unfold not in |- *; intros;
+    [ discriminate | elim (H n0 H0); simplify_eq H1; trivial ] ].
+
+Qed.
+                                   
+
+(* \subsubsection{Entiers positifs} *)
+
+Fixpoint eq_pos (p1 p2 : positive) {struct p2} : bool :=
+  match p1 with
+  | xI n1 => match p2 with
+             | xI n2 => eq_pos n1 n2
+             | _ => false
+             end
+  | xO n1 => match p2 with
+             | xO n2 => eq_pos n1 n2
+             | _ => false
+             end
+  | xH => match p2 with
+          | xH => true
+          | _ => false
+          end
+  end.
+
+Theorem eq_pos_true : forall t1 t2 : positive, eq_pos t1 t2 = true -> t1 = t2.
+
+simple induction t1;
+ [ intros p H t2; case t2;
+    [ simpl in |- *; intros; rewrite (H p0 H0); trivial
+    | absurd_case
+    | absurd_case ]
+ | intros p H t2; case t2;
+    [ absurd_case
+    | simpl in |- *; intros; rewrite (H p0 H0); trivial
+    | absurd_case ]
+ | intro t2; case t2; [ absurd_case | absurd_case | auto ] ].
+
+Qed.
+  
+Theorem eq_pos_false :
+ forall t1 t2 : positive, eq_pos t1 t2 = false -> t1 <> t2.
+
+simple induction t1;
+ [ intros p H t2; case t2;
+    [ simpl in |- *; unfold not in |- *; intros; elim (H p0 H0);
+       simplify_eq H1; auto
+    | trivial_case
+    | trivial_case ]
+ | intros p H t2; case t2;
+    [ trivial_case
+    | simpl in |- *; unfold not in |- *; intros; elim (H p0 H0);
+       simplify_eq H1; auto
+    | trivial_case ]
+ | intros t2; case t2; [ trivial_case | trivial_case | absurd_case ] ].
+Qed.
+
+(* \subsubsection{Entiers relatifs} *)
+
+Definition eq_Z (z1 z2 : Z) : bool :=
+  match z1 with
+  | Z0 => match z2 with
+          | Z0 => true
+          | _ => false
+          end
+  | Zpos p1 => match z2 with
+               | Zpos p2 => eq_pos p1 p2
+               | _ => false
+               end
+  | Zneg p1 => match z2 with
+               | Zneg p2 => eq_pos p1 p2
+               | _ => false
+               end
+  end.
+
+Theorem eq_Z_true : forall t1 t2 : Z, eq_Z t1 t2 = true -> t1 = t2.
+
+simple induction t1;
+ [ intros t2; case t2; [ auto | absurd_case | absurd_case ]
+ | intros p t2; case t2;
+    [ absurd_case
+    | simpl in |- *; intros; rewrite (eq_pos_true p p0 H); trivial
+    | absurd_case ]
+ | intros p t2; case t2;
+    [ absurd_case
+    | absurd_case
+    | simpl in |- *; intros; rewrite (eq_pos_true p p0 H); trivial ] ].
+
+Qed.
+
+Theorem eq_Z_false : forall t1 t2 : Z, eq_Z t1 t2 = false -> t1 <> t2.
+
+simple induction t1;
+ [ intros t2; case t2; [ absurd_case | trivial_case | trivial_case ]
+ | intros p t2; case t2;
+    [ absurd_case
+    | simpl in |- *; unfold not in |- *; intros; elim (eq_pos_false p p0 H);
+       simplify_eq H0; auto
+    | trivial_case ]
+ | intros p t2; case t2;
+    [ absurd_case
+    | trivial_case
+    | simpl in |- *; unfold not in |- *; intros; elim (eq_pos_false p p0 H);
+       simplify_eq H0; auto ] ].
+Qed.
+
+(* \subsubsection{Termes réifiés} *)
+
+Fixpoint eq_term (t1 t2 : term) {struct t2} : bool :=
+  match t1 with
+  | Tint st1 => match t2 with
+                | Tint st2 => eq_Z st1 st2
+                | _ => false
+                end
+  | Tplus st11 st12 =>
+      match t2 with
+      | Tplus st21 st22 => eq_term st11 st21 && eq_term st12 st22
+      | _ => false
+      end
+  | Tmult st11 st12 =>
+      match t2 with
+      | Tmult st21 st22 => eq_term st11 st21 && eq_term st12 st22
+      | _ => false
+      end
+  | Tminus st11 st12 =>
+      match t2 with
+      | Tminus st21 st22 => eq_term st11 st21 && eq_term st12 st22
+      | _ => false
+      end
+  | Topp st1 => match t2 with
+                | Topp st2 => eq_term st1 st2
+                | _ => false
+                end
+  | Tvar st1 => match t2 with
+                | Tvar st2 => eq_nat st1 st2
+                | _ => false
+                end
+  end.  
+
+Theorem eq_term_true : forall t1 t2 : term, eq_term t1 t2 = true -> t1 = t2.
+
+
+simple induction t1; intros until t2; case t2; try absurd_case; simpl in |- *;
+ [ intros; elim eq_Z_true with (1 := H); trivial
+ | intros t21 t22 H3; elim andb_prop with (1 := H3); intros H4 H5;
+    elim H with (1 := H4); elim H0 with (1 := H5); 
+    trivial
+ | intros t21 t22 H3; elim andb_prop with (1 := H3); intros H4 H5;
+    elim H with (1 := H4); elim H0 with (1 := H5); 
+    trivial
+ | intros t21 t22 H3; elim andb_prop with (1 := H3); intros H4 H5;
+    elim H with (1 := H4); elim H0 with (1 := H5); 
+    trivial
+ | intros t21 H3; elim H with (1 := H3); trivial
+ | intros; elim eq_nat_true with (1 := H); trivial ].
+
+Qed.
+
+Theorem eq_term_false :
+ forall t1 t2 : term, eq_term t1 t2 = false -> t1 <> t2.
+
+simple induction t1;
+ [ intros z t2; case t2; try trivial_case; simpl in |- *; unfold not in |- *;
+    intros; elim eq_Z_false with (1 := H); simplify_eq H0; 
+    auto
+ | intros t11 H1 t12 H2 t2; case t2; try trivial_case; simpl in |- *;
+    intros t21 t22 H3; unfold not in |- *; intro H4;
+    elim andb_false_elim with (1 := H3); intros H5;
+    [ elim H1 with (1 := H5); simplify_eq H4; auto
+    | elim H2 with (1 := H5); simplify_eq H4; auto ]
+ | intros t11 H1 t12 H2 t2; case t2; try trivial_case; simpl in |- *;
+    intros t21 t22 H3; unfold not in |- *; intro H4;
+    elim andb_false_elim with (1 := H3); intros H5;
+    [ elim H1 with (1 := H5); simplify_eq H4; auto
+    | elim H2 with (1 := H5); simplify_eq H4; auto ]
+ | intros t11 H1 t12 H2 t2; case t2; try trivial_case; simpl in |- *;
+    intros t21 t22 H3; unfold not in |- *; intro H4;
+    elim andb_false_elim with (1 := H3); intros H5;
+    [ elim H1 with (1 := H5); simplify_eq H4; auto
+    | elim H2 with (1 := H5); simplify_eq H4; auto ]
+ | intros t11 H1 t2; case t2; try trivial_case; simpl in |- *; intros t21 H3;
+    unfold not in |- *; intro H4; elim H1 with (1 := H3); 
+    simplify_eq H4; auto
+ | intros n t2; case t2; try trivial_case; simpl in |- *; unfold not in |- *;
+    intros; elim eq_nat_false with (1 := H); simplify_eq H0; 
+    auto ].
+
+Qed.
+
+(* \subsubsection{Tactiques pour éliminer ces tests} 
+
+   Si on se contente de faire un [Case (eq_typ t1 t2)] on perd 
+   totalement dans chaque branche le fait que [t1=t2] ou [~t1=t2].
+
+   Initialement, les développements avaient été réalisés avec les
+   tests rendus par [Decide Equality], c'est à dire un test rendant
+   des termes du type [{t1=t2}+{~t1=t2}]. Faire une élimination sur un 
+   tel test préserve bien l'information voulue mais calculatoirement de
+   telles fonctions sont trop lentes. *)
+
+(* Le théorème suivant permet de garder dans les hypothèses la valeur
+   du booléen lors de l'élimination. *)
+
+Theorem bool_ind2 :
+ forall (P : bool -> Prop) (b : bool),
+ (b = true -> P true) -> (b = false -> P false) -> P b.
+
+simple induction b; auto.
+Qed.
+
+(* Les tactiques définies si après se comportent exactement comme si on
+   avait utilisé le test précédent et fait une elimination dessus. *)
+
+Ltac elim_eq_term t1 t2 :=
+  pattern (eq_term t1 t2) in |- *; apply bool_ind2; intro Aux;
+   [ generalize (eq_term_true t1 t2 Aux); clear Aux
+   | generalize (eq_term_false t1 t2 Aux); clear Aux ].
+
+Ltac elim_eq_Z t1 t2 :=
+  pattern (eq_Z t1 t2) in |- *; apply bool_ind2; intro Aux;
+   [ generalize (eq_Z_true t1 t2 Aux); clear Aux
+   | generalize (eq_Z_false t1 t2 Aux); clear Aux ].
+
+Ltac elim_eq_pos t1 t2 :=
+  pattern (eq_pos t1 t2) in |- *; apply bool_ind2; intro Aux;
+   [ generalize (eq_pos_true t1 t2 Aux); clear Aux
+   | generalize (eq_pos_false t1 t2 Aux); clear Aux ].
+
+(* \subsubsection{Comparaison sur Z} *)
+
+(* Sujet très lié au précédent : on introduit la tactique d'élimination
+   avec son théorème *)
+
+Theorem relation_ind2 :
+ forall (P : Datatypes.comparison -> Prop) (b : Datatypes.comparison),
+ (b = Datatypes.Eq -> P Datatypes.Eq) ->
+ (b = Datatypes.Lt -> P Datatypes.Lt) ->
+ (b = Datatypes.Gt -> P Datatypes.Gt) -> P b.
+
+simple induction b; auto.
+Qed.
+
+Ltac elim_Zcompare t1 t2 := pattern (t1 ?= t2)%Z in |- *; apply relation_ind2.
+
+(* \subsection{Interprétations}
+   \subsubsection{Interprétation des termes dans Z} *)
+
+Fixpoint interp_term (env : list Z) (t : term) {struct t} : Z :=
+  match t with
+  | Tint x => x
+  | Tplus t1 t2 => (interp_term env t1 + interp_term env t2)%Z
+  | Tmult t1 t2 => (interp_term env t1 * interp_term env t2)%Z
+  | Tminus t1 t2 => (interp_term env t1 - interp_term env t2)%Z
+  | Topp t => (- interp_term env t)%Z
+  | Tvar n => nth n env 0%Z
+  end.
+
+(* \subsubsection{Interprétation des prédicats} *) 
+Fixpoint interp_proposition (envp : PropList) (env : list Z)
+ (p : proposition) {struct p} : Prop :=
+  match p with
+  | EqTerm t1 t2 => interp_term env t1 = interp_term env t2
+  | LeqTerm t1 t2 => (interp_term env t1 <= interp_term env t2)%Z
+  | TrueTerm => True
+  | FalseTerm => False
+  | Tnot p' => ~ interp_proposition envp env p'
+  | GeqTerm t1 t2 => (interp_term env t1 >= interp_term env t2)%Z
+  | GtTerm t1 t2 => (interp_term env t1 > interp_term env t2)%Z
+  | LtTerm t1 t2 => (interp_term env t1 < interp_term env t2)%Z
+  | NeqTerm t1 t2 => Zne (interp_term env t1) (interp_term env t2)
+  | Tor p1 p2 =>
+      interp_proposition envp env p1 \/ interp_proposition envp env p2
+  | Tand p1 p2 =>
+      interp_proposition envp env p1 /\ interp_proposition envp env p2
+  | Timp p1 p2 =>
+      interp_proposition envp env p1 -> interp_proposition envp env p2
+  | Tprop n => nthProp n envp True
+  end.
+
+(* \subsubsection{Inteprétation des listes d'hypothèses}
+   \paragraph{Sous forme de conjonction}
+   Interprétation sous forme d'une conjonction d'hypothèses plus faciles
+   à manipuler individuellement *)
+
+Fixpoint interp_hyps (envp : PropList) (env : list Z) 
+ (l : list proposition) {struct l} : Prop :=
+  match l with
+  | nil => True
+  | p' :: l' => interp_proposition envp env p' /\ interp_hyps envp env l'
+  end.
+
+(* \paragraph{sous forme de but}
+   C'est cette interpétation que l'on utilise sur le but (car on utilise
+   [Generalize] et qu'une conjonction est forcément lourde (répétition des
+   types dans les conjonctions intermédiaires) *)
+
+Fixpoint interp_goal_concl (envp : PropList) (env : list Z) 
+ (c : proposition) (l : list proposition) {struct l} : Prop :=
+  match l with
+  | nil => interp_proposition envp env c
+  | p' :: l' =>
+      interp_proposition envp env p' -> interp_goal_concl envp env c l'
+  end.
+
+Notation interp_goal :=
+  (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+   interp_goal_concl envp env FalseTerm l) (only parsing).
+
+(* Les théorèmes qui suivent assurent la correspondance entre les deux
+   interprétations. *)
+
+Theorem goal_to_hyps :
+ forall (envp : PropList) (env : list Z) (l : list proposition),
+ (interp_hyps envp env l -> False) ->
+ (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+  interp_goal_concl envp env FalseTerm l) envp env l.
+
+simple induction l;
+ [ simpl in |- *; auto
+ | simpl in |- *; intros a l1 H1 H2 H3; apply H1; intro H4; apply H2; auto ].
+Qed.
+
+Theorem hyps_to_goal :
+ forall (envp : PropList) (env : list Z) (l : list proposition),
+ (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+  interp_goal_concl envp env FalseTerm l) envp env l ->
+ interp_hyps envp env l -> False.
+
+simple induction l; simpl in |- *; [ auto | intros; apply H; elim H1; auto ].
+Qed.
+      
+(* \subsection{Manipulations sur les hypothèses} *)
+
+(* \subsubsection{Définitions de base de stabilité pour la réflexion} *)
+(* Une opération laisse un terme stable si l'égalité est préservée *)
+Definition term_stable (f : term -> term) :=
+  forall (e : list Z) (t : term), interp_term e t = interp_term e (f t).
+
+(* Une opération est valide sur une hypothèse, si l'hypothèse implique le
+   résultat de l'opération. \emph{Attention : cela ne concerne que des 
+   opérations sur les hypothèses et non sur les buts (contravariance)}.
+   On définit la validité pour une opération prenant une ou deux propositions
+   en argument (cela suffit pour omega). *)
+
+Definition valid1 (f : proposition -> proposition) :=
+  forall (ep : PropList) (e : list Z) (p1 : proposition),
+  interp_proposition ep e p1 -> interp_proposition ep e (f p1).
+
+Definition valid2 (f : proposition -> proposition -> proposition) :=
+  forall (ep : PropList) (e : list Z) (p1 p2 : proposition),
+  interp_proposition ep e p1 ->
+  interp_proposition ep e p2 -> interp_proposition ep e (f p1 p2).
+
+(* Dans cette notion de validité, la fonction prend directement une 
+   liste de propositions et rend une nouvelle liste de proposition. 
+   On reste contravariant *)
+
+Definition valid_hyps (f : list proposition -> list proposition) :=
+  forall (ep : PropList) (e : list Z) (lp : list proposition),
+  interp_hyps ep e lp -> interp_hyps ep e (f lp).
+
+(* Enfin ce théorème élimine la contravariance et nous ramène à une 
+   opération sur les buts *)
+
+ Theorem valid_goal :
+  forall (ep : PropList) (env : list Z) (l : list proposition)
+    (a : list proposition -> list proposition),
+  valid_hyps a ->
+  (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+   interp_goal_concl envp env FalseTerm l) ep env (
+    a l) ->
+  (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+   interp_goal_concl envp env FalseTerm l) ep env l.
+
+intros; simpl in |- *; apply goal_to_hyps; intro H1;
+ apply (hyps_to_goal ep env (a l) H0); apply H; assumption.
+Qed.
+
+(* \subsubsection{Généralisation a des listes de buts (disjonctions)} *)
+
+
+Fixpoint interp_list_hyps (envp : PropList) (env : list Z)
+ (l : list (list proposition)) {struct l} : Prop :=
+  match l with
+  | nil => False
+  | h :: l' => interp_hyps envp env h \/ interp_list_hyps envp env l'
+  end.
+
+Fixpoint interp_list_goal (envp : PropList) (env : list Z)
+ (l : list (list proposition)) {struct l} : Prop :=
+  match l with
+  | nil => True
+  | h :: l' =>
+      (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+       interp_goal_concl envp env FalseTerm l) envp env h /\
+      interp_list_goal envp env l'
+  end.
+
+Theorem list_goal_to_hyps :
+ forall (envp : PropList) (env : list Z) (l : list (list proposition)),
+ (interp_list_hyps envp env l -> False) -> interp_list_goal envp env l.
+
+simple induction l; simpl in |- *;
+ [ auto
+ | intros h1 l1 H H1; split;
+    [ apply goal_to_hyps; intro H2; apply H1; auto
+    | apply H; intro H2; apply H1; auto ] ].
+Qed.
+
+Theorem list_hyps_to_goal :
+ forall (envp : PropList) (env : list Z) (l : list (list proposition)),
+ interp_list_goal envp env l -> interp_list_hyps envp env l -> False.
+
+simple induction l; simpl in |- *;
+ [ auto
+ | intros h1 l1 H (H1, H2) H3; elim H3; intro H4;
+    [ apply hyps_to_goal with (1 := H1); assumption | auto ] ].
+Qed.
+
+Definition valid_list_hyps
+  (f : list proposition -> list (list proposition)) :=
+  forall (ep : PropList) (e : list Z) (lp : list proposition),
+  interp_hyps ep e lp -> interp_list_hyps ep e (f lp).
+
+Definition valid_list_goal
+  (f : list proposition -> list (list proposition)) :=
+  forall (ep : PropList) (e : list Z) (lp : list proposition),
+  interp_list_goal ep e (f lp) ->
+  (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+   interp_goal_concl envp env FalseTerm l) ep e lp.
+
+Theorem goal_valid :
+ forall f : list proposition -> list (list proposition),
+ valid_list_hyps f -> valid_list_goal f.
+
+unfold valid_list_goal in |- *; intros f H ep e lp H1; apply goal_to_hyps;
+ intro H2; apply list_hyps_to_goal with (1 := H1); 
+ apply (H ep e lp); assumption.
+Qed.
+ 
+Theorem append_valid :
+ forall (ep : PropList) (e : list Z) (l1 l2 : list (list proposition)),
+ interp_list_hyps ep e l1 \/ interp_list_hyps ep e l2 ->
+ interp_list_hyps ep e (l1 ++ l2).
+
+intros ep e; simple induction l1;
+ [ simpl in |- *; intros l2 [H| H]; [ contradiction | trivial ]
+ | simpl in |- *; intros h1 t1 HR l2 [[H| H]| H];
+    [ auto
+    | right; apply (HR l2); left; trivial
+    | right; apply (HR l2); right; trivial ] ].
+
+Qed.
+
+(* \subsubsection{Opérateurs valides sur les hypothèses} *)
+
+(* Extraire une hypothèse de la liste *)
+Definition nth_hyps (n : nat) (l : list proposition) := nth n l TrueTerm.
+
+Theorem nth_valid :
+ forall (ep : PropList) (e : list Z) (i : nat) (l : list proposition),
+ interp_hyps ep e l -> interp_proposition ep e (nth_hyps i l).
+
+unfold nth_hyps in |- *; simple induction i;
+ [ simple induction l; simpl in |- *; [ auto | intros; elim H0; auto ]
+ | intros n H; simple induction l;
+    [ simpl in |- *; trivial
+    | intros; simpl in |- *; apply H; elim H1; auto ] ].
+Qed.
+
+(* Appliquer une opération (valide) sur deux hypothèses extraites de 
+   la liste et ajouter le résultat à la liste. *)
+Definition apply_oper_2 (i j : nat)
+  (f : proposition -> proposition -> proposition) (l : list proposition) :=
+  f (nth_hyps i l) (nth_hyps j l) :: l.
+
+Theorem apply_oper_2_valid :
+ forall (i j : nat) (f : proposition -> proposition -> proposition),
+ valid2 f -> valid_hyps (apply_oper_2 i j f).
+
+intros i j f Hf; unfold apply_oper_2, valid_hyps in |- *; simpl in |- *;
+ intros lp Hlp; split; [ apply Hf; apply nth_valid; assumption | assumption ].
+Qed.
+
+(* Modifier une hypothèse par application d'une opération valide *)
+
+Fixpoint apply_oper_1 (i : nat) (f : proposition -> proposition)
+ (l : list proposition) {struct i} : list proposition :=
+  match l with
+  | nil => nil (A:=proposition)
+  | p :: l' =>
+      match i with
+      | O => f p :: l'
+      | S j => p :: apply_oper_1 j f l'
+      end
+  end.
+
+Theorem apply_oper_1_valid :
+ forall (i : nat) (f : proposition -> proposition),
+ valid1 f -> valid_hyps (apply_oper_1 i f).
+
+unfold valid_hyps in |- *; intros i f Hf ep e; elim i;
+ [ intro lp; case lp;
+    [ simpl in |- *; trivial
+    | simpl in |- *; intros p l' (H1, H2); split;
+       [ apply Hf with (1 := H1) | assumption ] ]
+ | intros n Hrec lp; case lp;
+    [ simpl in |- *; auto
+    | simpl in |- *; intros p l' (H1, H2); split;
+       [ assumption | apply Hrec; assumption ] ] ].
+
+Qed.
+
+(* \subsubsection{Manipulations de termes} *)
+(* Les fonctions suivantes permettent d'appliquer une fonction de
+   réécriture sur un sous terme du terme principal. Avec la composition,
+   cela permet de construire des réécritures complexes proches des 
+   tactiques de conversion *)
+
+Definition apply_left (f : term -> term) (t : term) :=
+  match t with
+  | Tplus x y => Tplus (f x) y
+  | Tmult x y => Tmult (f x) y
+  | Topp x => Topp (f x)
+  | x => x
+  end.
+
+Definition apply_right (f : term -> term) (t : term) :=
+  match t with
+  | Tplus x y => Tplus x (f y)
+  | Tmult x y => Tmult x (f y)
+  | x => x
+  end.
+
+Definition apply_both (f g : term -> term) (t : term) :=
+  match t with
+  | Tplus x y => Tplus (f x) (g y)
+  | Tmult x y => Tmult (f x) (g y)
+  | x => x
+  end.
+
+(* Les théorèmes suivants montrent la stabilité (conditionnée) des 
+   fonctions. *)
+
+Theorem apply_left_stable :
+ forall f : term -> term, term_stable f -> term_stable (apply_left f).
+
+unfold term_stable in |- *; intros f H e t; case t; auto; simpl in |- *;
+ intros; elim H; trivial.
+Qed.
+
+Theorem apply_right_stable :
+ forall f : term -> term, term_stable f -> term_stable (apply_right f).
+
+unfold term_stable in |- *; intros f H e t; case t; auto; simpl in |- *;
+ intros t0 t1; elim H; trivial.
+Qed.
+
+Theorem apply_both_stable :
+ forall f g : term -> term,
+ term_stable f -> term_stable g -> term_stable (apply_both f g).
+
+unfold term_stable in |- *; intros f g H1 H2 e t; case t; auto; simpl in |- *;
+ intros t0 t1; elim H1; elim H2; trivial.
+Qed.
+
+Theorem compose_term_stable :
+ forall f g : term -> term,
+ term_stable f -> term_stable g -> term_stable (fun t : term => f (g t)).
+
+unfold term_stable in |- *; intros f g Hf Hg e t; elim Hf; apply Hg.
+Qed.
+
+(* \subsection{Les règles de réécriture} *)
+(* Chacune des règles de réécriture est accompagnée par sa preuve de 
+   stabilité. Toutes ces preuves ont la même forme : il faut analyser 
+   suivant la forme du terme (élimination de chaque Case). On a besoin d'une
+   élimination uniquement dans les cas d'utilisation d'égalité décidable. 
+
+   Cette tactique itère la décomposition des Case. Elle est
+   constituée de deux fonctions s'appelant mutuellement :
+   \begin{itemize} 
+    \item une fonction d'enrobage qui lance la recherche sur le but,
+    \item une fonction récursive qui décompose ce but. Quand elle a trouvé un
+          Case, elle l'élimine. 
+   \end{itemize}  
+   Les motifs sur les cas sont très imparfaits et dans certains cas, il
+   semble que cela ne marche pas. On aimerait plutot un motif de la
+   forme [ Case (?1 :: T) of _ end ] permettant de s'assurer que l'on 
+   utilise le bon type.
+
+   Chaque élimination introduit correctement exactement le nombre d'hypothèses
+   nécessaires et conserve dans le cas d'une égalité la connaissance du
+   résultat du test en faisant la réécriture. Pour un test de comparaison,
+   on conserve simplement le résultat.
+
+   Cette fonction déborde très largement la résolution des réécritures
+   simples et fait une bonne partie des preuves des pas de Omega.
+*)
+
+(* \subsubsection{La tactique pour prouver la stabilité} *)
+
+Ltac loop t :=
+  match constr:t with
+  | (?X1 = ?X2) => 
+                    (* Global *)
+                    loop X1 || loop X2
+  | (_ -> ?X1) => loop X1
+  | (interp_hyps _ _ ?X1) =>
+      
+  (* Interpretations *)
+  loop X1
+  | (interp_list_hyps _ _ ?X1) => loop X1
+  | (interp_proposition _ _ ?X1) => loop X1
+  | (interp_term _ ?X1) => loop X1
+  | (EqTerm ?X1 ?X2) =>
+      
+       (* Propositions *)
+       loop X1 || loop X2
+  | (LeqTerm ?X1 ?X2) => loop X1 || loop X2
+  | (Tplus ?X1 ?X2) => 
+                        (* Termes *)
+                        loop X1 || loop X2
+  | (Tminus ?X1 ?X2) => loop X1 || loop X2
+  | (Tmult ?X1 ?X2) => loop X1 || loop X2
+  | (Topp ?X1) => loop X1
+  | (Tint ?X1) =>
+      loop X1
+  | match ?X1 with
+    | EqTerm x x0 => _
+    | LeqTerm x x0 => _
+    | TrueTerm => _
+    | FalseTerm => _
+    | Tnot x => _
+    | GeqTerm x x0 => _
+    | GtTerm x x0 => _
+    | LtTerm x x0 => _
+    | NeqTerm x x0 => _
+    | Tor x x0 => _
+    | Tand x x0 => _
+    | Timp x x0 => _
+    | Tprop x => _
+    end =>
+      
+       (* Eliminations *)
+       case X1;
+       [ intro; intro
+       | intro; intro
+       | idtac
+       | idtac
+       | intro
+       | intro; intro
+       | intro; intro
+       | intro; intro
+       | intro; intro
+       | intro; intro
+       | intro; intro
+       | intro; intro
+       | intro ]; auto; Simplify
+  | match ?X1 with
+    | Tint x => _
+    | Tplus x x0 => _
+    | Tmult x x0 => _
+    | Tminus x x0 => _
+    | Topp x => _
+    | Tvar x => _
+    end =>
+      case X1;
+       [ intro | intro; intro | intro; intro | intro; intro | intro | intro ];
+       auto; Simplify
+  | match (?X1 ?= ?X2)%Z with
+    | Datatypes.Eq => _
+    | Datatypes.Lt => _
+    | Datatypes.Gt => _
+    end =>
+      elim_Zcompare X1 X2; intro; auto; Simplify
+  | match ?X1 with
+    | Z0 => _
+    | Zpos x => _
+    | Zneg x => _
+    end =>
+      case X1; [ idtac | intro | intro ]; auto; Simplify
+  | (if eq_Z ?X1 ?X2 then _ else _) =>
+      elim_eq_Z X1 X2; intro H; [ rewrite H; clear H | clear H ];
+       simpl in |- *; auto; Simplify
+  | (if eq_term ?X1 ?X2 then _ else _) =>
+      elim_eq_term X1 X2; intro H; [ rewrite H; clear H | clear H ];
+       simpl in |- *; auto; Simplify
+  | (if eq_pos ?X1 ?X2 then _ else _) =>
+      elim_eq_pos X1 X2; intro H; [ rewrite H; clear H | clear H ];
+       simpl in |- *; auto; Simplify
+  | _ => fail
+  end
+ with Simplify := match goal with
+                  |  |- ?X1 => try loop X1
+                  | _ => idtac
+                  end.
+
+
+Ltac prove_stable x th :=
+  match constr:x with
+  | ?X1 =>
+      unfold term_stable, X1 in |- *; intros; Simplify; simpl in |- *;
+       apply th
+  end.
+
+(* \subsubsection{Les règles elle mêmes} *)
+Definition Tplus_assoc_l (t : term) :=
+  match t with
+  | Tplus n (Tplus m p) => Tplus (Tplus n m) p
+  | _ => t
+  end.
+
+Theorem Tplus_assoc_l_stable : term_stable Tplus_assoc_l.
+
+prove_stable Tplus_assoc_l Zplus_assoc.
+Qed.
+
+Definition Tplus_assoc_r (t : term) :=
+  match t with
+  | Tplus (Tplus n m) p => Tplus n (Tplus m p)
+  | _ => t
+  end.
+
+Theorem Tplus_assoc_r_stable : term_stable Tplus_assoc_r.
+
+prove_stable Tplus_assoc_r Zplus_assoc_reverse.
+Qed.
+
+Definition Tmult_assoc_r (t : term) :=
+  match t with
+  | Tmult (Tmult n m) p => Tmult n (Tmult m p)
+  | _ => t
+  end.
+
+Theorem Tmult_assoc_r_stable : term_stable Tmult_assoc_r.
+
+prove_stable Tmult_assoc_r Zmult_assoc_reverse.
+Qed.
+
+Definition Tplus_permute (t : term) :=
+  match t with
+  | Tplus n (Tplus m p) => Tplus m (Tplus n p)
+  | _ => t
+  end.
+
+Theorem Tplus_permute_stable : term_stable Tplus_permute.
+
+prove_stable Tplus_permute Zplus_permute.
+Qed.
+
+Definition Tplus_sym (t : term) :=
+  match t with
+  | Tplus x y => Tplus y x
+  | _ => t
+  end.
+
+Theorem Tplus_sym_stable : term_stable Tplus_sym.
+
+prove_stable Tplus_sym Zplus_comm.
+Qed.
+
+Definition Tmult_sym (t : term) :=
+  match t with
+  | Tmult x y => Tmult y x
+  | _ => t
+  end.
+
+Theorem Tmult_sym_stable : term_stable Tmult_sym.
+
+prove_stable Tmult_sym Zmult_comm.
+Qed.
+
+Definition T_OMEGA10 (t : term) :=
+  match t with
+  | Tplus (Tmult (Tplus (Tmult v (Tint c1)) l1) (Tint k1)) (Tmult (Tplus
+    (Tmult v' (Tint c2)) l2) (Tint k2)) =>
+      match eq_term v v' with
+      | true =>
+          Tplus (Tmult v (Tint (c1 * k1 + c2 * k2)))
+            (Tplus (Tmult l1 (Tint k1)) (Tmult l2 (Tint k2)))
+      | false => t
+      end
+  | _ => t
+  end.
+
+Theorem T_OMEGA10_stable : term_stable T_OMEGA10.
+
+prove_stable T_OMEGA10 OMEGA10.
+Qed.
+
+Definition T_OMEGA11 (t : term) :=
+  match t with
+  | Tplus (Tmult (Tplus (Tmult v1 (Tint c1)) l1) (Tint k1)) l2 =>
+      Tplus (Tmult v1 (Tint (c1 * k1))) (Tplus (Tmult l1 (Tint k1)) l2)
+  | _ => t
+  end.
+
+Theorem T_OMEGA11_stable : term_stable T_OMEGA11.
+
+prove_stable T_OMEGA11 OMEGA11.
+Qed.
+
+Definition T_OMEGA12 (t : term) :=
+  match t with
+  | Tplus l1 (Tmult (Tplus (Tmult v2 (Tint c2)) l2) (Tint k2)) =>
+      Tplus (Tmult v2 (Tint (c2 * k2))) (Tplus l1 (Tmult l2 (Tint k2)))
+  | _ => t
+  end.
+
+Theorem T_OMEGA12_stable : term_stable T_OMEGA12.
+
+prove_stable T_OMEGA12 OMEGA12.
+Qed. 
+
+Definition T_OMEGA13 (t : term) :=
+  match t with
+  | Tplus (Tplus (Tmult v (Tint (Zpos x))) l1) (Tplus (Tmult v' (Tint (Zneg
+    x'))) l2) =>
+      match eq_term v v' with
+      | true => match eq_pos x x' with
+                | true => Tplus l1 l2
+                | false => t
+                end
+      | false => t
+      end
+  | Tplus (Tplus (Tmult v (Tint (Zneg x))) l1) (Tplus (Tmult v' (Tint (Zpos
+    x'))) l2) =>
+      match eq_term v v' with
+      | true => match eq_pos x x' with
+                | true => Tplus l1 l2
+                | false => t
+                end
+      | false => t
+      end
+  | _ => t
+  end.
+
+Theorem T_OMEGA13_stable : term_stable T_OMEGA13.
+
+unfold term_stable, T_OMEGA13 in |- *; intros; Simplify; simpl in |- *;
+ [ apply OMEGA13 | apply OMEGA14 ].
+Qed.
+
+Definition T_OMEGA15 (t : term) :=
+  match t with
+  | Tplus (Tplus (Tmult v (Tint c1)) l1) (Tmult (Tplus (Tmult v' (Tint c2))
+    l2) (Tint k2)) =>
+      match eq_term v v' with
+      | true =>
+          Tplus (Tmult v (Tint (c1 + c2 * k2)))
+            (Tplus l1 (Tmult l2 (Tint k2)))
+      | false => t
+      end
+  | _ => t
+  end.
+
+Theorem T_OMEGA15_stable : term_stable T_OMEGA15.
+
+prove_stable T_OMEGA15 OMEGA15.
+Qed.
+
+Definition T_OMEGA16 (t : term) :=
+  match t with
+  | Tmult (Tplus (Tmult v (Tint c)) l) (Tint k) =>
+      Tplus (Tmult v (Tint (c * k))) (Tmult l (Tint k))
+  | _ => t
+  end.
+
+
+Theorem T_OMEGA16_stable : term_stable T_OMEGA16.
+
+prove_stable T_OMEGA16 OMEGA16.
+Qed.
+
+Definition Tred_factor5 (t : term) :=
+  match t with
+  | Tplus (Tmult x (Tint Z0)) y => y
+  | _ => t
+  end.
+
+Theorem Tred_factor5_stable : term_stable Tred_factor5.
+
+
+prove_stable Tred_factor5 Zred_factor5.
+Qed.
+
+Definition Topp_plus (t : term) :=
+  match t with
+  | Topp (Tplus x y) => Tplus (Topp x) (Topp y)
+  | _ => t
+  end.
+
+Theorem Topp_plus_stable : term_stable Topp_plus.
+
+prove_stable Topp_plus Zopp_plus_distr.
+Qed.
+
+
+Definition Topp_opp (t : term) :=
+  match t with
+  | Topp (Topp x) => x
+  | _ => t
+  end.
+
+Theorem Topp_opp_stable : term_stable Topp_opp.
+
+prove_stable Topp_opp Zopp_involutive.
+Qed.
+
+Definition Topp_mult_r (t : term) :=
+  match t with
+  | Topp (Tmult x (Tint k)) => Tmult x (Tint (- k))
+  | _ => t
+  end.
+
+Theorem Topp_mult_r_stable : term_stable Topp_mult_r.
+
+prove_stable Topp_mult_r Zopp_mult_distr_r.
+Qed.
+
+Definition Topp_one (t : term) :=
+  match t with
+  | Topp x => Tmult x (Tint (-1))
+  | _ => t
+  end.
+
+Theorem Topp_one_stable : term_stable Topp_one.
+
+prove_stable Topp_one Zopp_eq_mult_neg_1.
+Qed.
+
+Definition Tmult_plus_distr (t : term) :=
+  match t with
+  | Tmult (Tplus n m) p => Tplus (Tmult n p) (Tmult m p)
+  | _ => t
+  end.
+
+Theorem Tmult_plus_distr_stable : term_stable Tmult_plus_distr.
+
+prove_stable Tmult_plus_distr Zmult_plus_distr_l.
+Qed.
+
+Definition Tmult_opp_left (t : term) :=
+  match t with
+  | Tmult (Topp x) (Tint y) => Tmult x (Tint (- y))
+  | _ => t
+  end.
+
+Theorem Tmult_opp_left_stable : term_stable Tmult_opp_left.
+
+prove_stable Tmult_opp_left Zmult_opp_comm.
+Qed.
+
+Definition Tmult_assoc_reduced (t : term) :=
+  match t with
+  | Tmult (Tmult n (Tint m)) (Tint p) => Tmult n (Tint (m * p))
+  | _ => t
+  end.
+
+Theorem Tmult_assoc_reduced_stable : term_stable Tmult_assoc_reduced.
+
+prove_stable Tmult_assoc_reduced Zmult_assoc_reverse.
+Qed.
+
+Definition Tred_factor0 (t : term) := Tmult t (Tint 1).
+
+Theorem Tred_factor0_stable : term_stable Tred_factor0.
+
+prove_stable Tred_factor0 Zred_factor0.
+Qed.
+
+Definition Tred_factor1 (t : term) :=
+  match t with
+  | Tplus x y =>
+      match eq_term x y with
+      | true => Tmult x (Tint 2)
+      | false => t
+      end
+  | _ => t
+  end.
+
+Theorem Tred_factor1_stable : term_stable Tred_factor1.
+
+prove_stable Tred_factor1 Zred_factor1.
+Qed.
+
+Definition Tred_factor2 (t : term) :=
+  match t with
+  | Tplus x (Tmult y (Tint k)) =>
+      match eq_term x y with
+      | true => Tmult x (Tint (1 + k))
+      | false => t
+      end
+  | _ => t
+  end.
+
+(* Attention : il faut rendre opaque [Zplus] pour éviter que la tactique
+   de simplification n'aille trop loin et défasse [Zplus 1 k] *)
+
+Opaque Zplus.
+
+Theorem Tred_factor2_stable : term_stable Tred_factor2.
+prove_stable Tred_factor2 Zred_factor2.
+Qed.
+
+Definition Tred_factor3 (t : term) :=
+  match t with
+  | Tplus (Tmult x (Tint k)) y =>
+      match eq_term x y with
+      | true => Tmult x (Tint (1 + k))
+      | false => t
+      end
+  | _ => t
+  end.
+
+Theorem Tred_factor3_stable : term_stable Tred_factor3.
+
+prove_stable Tred_factor3 Zred_factor3.
+Qed.
+
+
+Definition Tred_factor4 (t : term) :=
+  match t with
+  | Tplus (Tmult x (Tint k1)) (Tmult y (Tint k2)) =>
+      match eq_term x y with
+      | true => Tmult x (Tint (k1 + k2))
+      | false => t
+      end
+  | _ => t
+  end.
+
+Theorem Tred_factor4_stable : term_stable Tred_factor4.
+
+prove_stable Tred_factor4 Zred_factor4.
+Qed.
+
+Definition Tred_factor6 (t : term) := Tplus t (Tint 0).
+
+Theorem Tred_factor6_stable : term_stable Tred_factor6.
+
+prove_stable Tred_factor6 Zred_factor6.
+Qed.
+
+Transparent Zplus.
+
+Definition Tminus_def (t : term) :=
+  match t with
+  | Tminus x y => Tplus x (Topp y)
+  | _ => t
+  end.
+
+Theorem Tminus_def_stable : term_stable Tminus_def.
+
+(* Le théorème ne sert à rien. Le but est prouvé avant. *)
+prove_stable Tminus_def False.
+Qed.
+
+(* \subsection{Fonctions de réécriture complexes} *)
+
+(* \subsubsection{Fonction de réduction} *)
+(* Cette fonction réduit un terme dont la forme normale est un entier. Il
+   suffit pour cela d'échanger le constructeur [Tint] avec les opérateurs
+   réifiés. La réduction est ``gratuite''.  *)
+
+Fixpoint reduce (t : term) : term :=
+  match t with
+  | Tplus x y =>
+      match reduce x with
+      | Tint x' =>
+          match reduce y with
+          | Tint y' => Tint (x' + y')
+          | y' => Tplus (Tint x') y'
+          end
+      | x' => Tplus x' (reduce y)
+      end
+  | Tmult x y =>
+      match reduce x with
+      | Tint x' =>
+          match reduce y with
+          | Tint y' => Tint (x' * y')
+          | y' => Tmult (Tint x') y'
+          end
+      | x' => Tmult x' (reduce y)
+      end
+  | Tminus x y =>
+      match reduce x with
+      | Tint x' =>
+          match reduce y with
+          | Tint y' => Tint (x' - y')
+          | y' => Tminus (Tint x') y'
+          end
+      | x' => Tminus x' (reduce y)
+      end
+  | Topp x =>
+      match reduce x with
+      | Tint x' => Tint (- x')
+      | x' => Topp x'
+      end
+  | _ => t
+  end.
+
+Theorem reduce_stable : term_stable reduce.
+
+unfold term_stable in |- *; intros e t; elim t; auto;
+ try
+  (intros t0 H0 t1 H1; simpl in |- *; rewrite H0; rewrite H1;
+    (case (reduce t0);
+      [ intro z0; case (reduce t1); intros; auto
+      | intros; auto
+      | intros; auto
+      | intros; auto
+      | intros; auto
+      | intros; auto ])); intros t0 H0; simpl in |- *; 
+ rewrite H0; case (reduce t0); intros; auto.
+Qed.
+
+(* \subsubsection{Fusions}
+    \paragraph{Fusion de deux équations} *)
+(* On donne une somme de deux équations qui sont supposées normalisées. 
+   Cette fonction prend une trace de fusion en argument et transforme
+   le terme en une équation normalisée. C'est une version très simplifiée
+   du moteur de réécriture [rewrite]. *)
+
+Fixpoint fusion (trace : list t_fusion) (t : term) {struct trace} : term :=
+  match trace with
+  | nil => reduce t
+  | step :: trace' =>
+      match step with
+      | F_equal => apply_right (fusion trace') (T_OMEGA10 t)
+      | F_cancel => fusion trace' (Tred_factor5 (T_OMEGA10 t))
+      | F_left => apply_right (fusion trace') (T_OMEGA11 t)
+      | F_right => apply_right (fusion trace') (T_OMEGA12 t)
+      end
+  end.
+    
+Theorem fusion_stable : forall t : list t_fusion, term_stable (fusion t).
+
+simple induction t; simpl in |- *;
+ [ exact reduce_stable
+ | intros stp l H; case stp;
+    [ apply compose_term_stable;
+       [ apply apply_right_stable; assumption | exact T_OMEGA10_stable ]
+    | unfold term_stable in |- *; intros e t1; rewrite T_OMEGA10_stable;
+       rewrite Tred_factor5_stable; apply H
+    | apply compose_term_stable;
+       [ apply apply_right_stable; assumption | exact T_OMEGA11_stable ]
+    | apply compose_term_stable;
+       [ apply apply_right_stable; assumption | exact T_OMEGA12_stable ] ] ].
+
+Qed.
+
+(* \paragraph{Fusion de deux équations dont une sans coefficient} *)
+
+Definition fusion_right (trace : list t_fusion) (t : term) : term :=
+  match trace with
+  | nil => reduce t (* Il faut mettre un compute *)
+  | step :: trace' =>
+      match step with
+      | F_equal => apply_right (fusion trace') (T_OMEGA15 t)
+      | F_cancel => fusion trace' (Tred_factor5 (T_OMEGA15 t))
+      | F_left => apply_right (fusion trace') (Tplus_assoc_r t)
+      | F_right => apply_right (fusion trace') (T_OMEGA12 t)
+      end
+  end.
+
+(* \paragraph{Fusion avec anihilation} *)
+(* Normalement le résultat est une constante *)
+
+Fixpoint fusion_cancel (trace : nat) (t : term) {struct trace} : term :=
+  match trace with
+  | O => reduce t
+  | S trace' => fusion_cancel trace' (T_OMEGA13 t)
+  end.
+
+Theorem fusion_cancel_stable : forall t : nat, term_stable (fusion_cancel t).
+
+unfold term_stable, fusion_cancel in |- *; intros trace e; elim trace;
+ [ exact (reduce_stable e)
+ | intros n H t; elim H; exact (T_OMEGA13_stable e t) ].
+Qed. 
+
+(* \subsubsection{Opérations afines sur une équation} *)
+(* \paragraph{Multiplication scalaire et somme d'une constante} *)
+
+Fixpoint scalar_norm_add (trace : nat) (t : term) {struct trace} : term :=
+  match trace with
+  | O => reduce t
+  | S trace' => apply_right (scalar_norm_add trace') (T_OMEGA11 t)
+  end.
+
+Theorem scalar_norm_add_stable :
+ forall t : nat, term_stable (scalar_norm_add t).
+
+unfold term_stable, scalar_norm_add in |- *; intros trace; elim trace;
+ [ exact reduce_stable
+ | intros n H e t; elim apply_right_stable;
+    [ exact (T_OMEGA11_stable e t) | exact H ] ].
+Qed.
+   
+(* \paragraph{Multiplication scalaire} *)
+Fixpoint scalar_norm (trace : nat) (t : term) {struct trace} : term :=
+  match trace with
+  | O => reduce t
+  | S trace' => apply_right (scalar_norm trace') (T_OMEGA16 t)
+  end.
+
+Theorem scalar_norm_stable : forall t : nat, term_stable (scalar_norm t).
+
+unfold term_stable, scalar_norm in |- *; intros trace; elim trace;
+ [ exact reduce_stable
+ | intros n H e t; elim apply_right_stable;
+    [ exact (T_OMEGA16_stable e t) | exact H ] ].
+Qed.
+
+(* \paragraph{Somme d'une constante} *)
+Fixpoint add_norm (trace : nat) (t : term) {struct trace} : term :=
+  match trace with
+  | O => reduce t
+  | S trace' => apply_right (add_norm trace') (Tplus_assoc_r t)
+  end.
+
+Theorem add_norm_stable : forall t : nat, term_stable (add_norm t).
+
+unfold term_stable, add_norm in |- *; intros trace; elim trace;
+ [ exact reduce_stable
+ | intros n H e t; elim apply_right_stable;
+    [ exact (Tplus_assoc_r_stable e t) | exact H ] ].
+Qed.
+
+(* \subsection{La fonction de normalisation des termes (moteur de réécriture)} *)
+
+
+Fixpoint rewrite (s : step) : term -> term :=
+  match s with
+  | C_DO_BOTH s1 s2 => apply_both (rewrite s1) (rewrite s2)
+  | C_LEFT s => apply_left (rewrite s)
+  | C_RIGHT s => apply_right (rewrite s)
+  | C_SEQ s1 s2 => fun t : term => rewrite s2 (rewrite s1 t)
+  | C_NOP => fun t : term => t
+  | C_OPP_PLUS => Topp_plus
+  | C_OPP_OPP => Topp_opp
+  | C_OPP_MULT_R => Topp_mult_r
+  | C_OPP_ONE => Topp_one
+  | C_REDUCE => reduce
+  | C_MULT_PLUS_DISTR => Tmult_plus_distr
+  | C_MULT_OPP_LEFT => Tmult_opp_left
+  | C_MULT_ASSOC_R => Tmult_assoc_r
+  | C_PLUS_ASSOC_R => Tplus_assoc_r
+  | C_PLUS_ASSOC_L => Tplus_assoc_l
+  | C_PLUS_PERMUTE => Tplus_permute
+  | C_PLUS_SYM => Tplus_sym
+  | C_RED0 => Tred_factor0
+  | C_RED1 => Tred_factor1
+  | C_RED2 => Tred_factor2
+  | C_RED3 => Tred_factor3
+  | C_RED4 => Tred_factor4
+  | C_RED5 => Tred_factor5
+  | C_RED6 => Tred_factor6
+  | C_MULT_ASSOC_REDUCED => Tmult_assoc_reduced
+  | C_MINUS => Tminus_def
+  | C_MULT_SYM => Tmult_sym
+  end.
+
+Theorem rewrite_stable : forall s : step, term_stable (rewrite s).
+
+simple induction s; simpl in |- *;
+ [ intros; apply apply_both_stable; auto
+ | intros; apply apply_left_stable; auto
+ | intros; apply apply_right_stable; auto
+ | unfold term_stable in |- *; intros; elim H0; apply H
+ | unfold term_stable in |- *; auto
+ | exact Topp_plus_stable
+ | exact Topp_opp_stable
+ | exact Topp_mult_r_stable
+ | exact Topp_one_stable
+ | exact reduce_stable
+ | exact Tmult_plus_distr_stable
+ | exact Tmult_opp_left_stable
+ | exact Tmult_assoc_r_stable
+ | exact Tplus_assoc_r_stable
+ | exact Tplus_assoc_l_stable
+ | exact Tplus_permute_stable
+ | exact Tplus_sym_stable
+ | exact Tred_factor0_stable
+ | exact Tred_factor1_stable
+ | exact Tred_factor2_stable
+ | exact Tred_factor3_stable
+ | exact Tred_factor4_stable
+ | exact Tred_factor5_stable
+ | exact Tred_factor6_stable
+ | exact Tmult_assoc_reduced_stable
+ | exact Tminus_def_stable
+ | exact Tmult_sym_stable ].
+Qed.
+
+(* \subsection{tactiques de résolution d'un but omega normalisé} 
+   Trace de la procédure 
+\subsubsection{Tactiques générant une contradiction}
+\paragraph{[O_CONSTANT_NOT_NUL]} *)
+
+Definition constant_not_nul (i : nat) (h : list proposition) :=
+  match nth_hyps i h with
+  | EqTerm (Tint Z0) (Tint n) =>
+      match eq_Z n 0 with
+      | true => h
+      | false => absurd
+      end
+  | _ => h
+  end.
+
+Theorem constant_not_nul_valid :
+ forall i : nat, valid_hyps (constant_not_nul i).
+
+unfold valid_hyps, constant_not_nul in |- *; intros;
+ generalize (nth_valid ep e i lp); Simplify; simpl in |- *;
+ elim_eq_Z ipattern:z0 0%Z; auto; simpl in |- *; intros H1 H2; 
+ elim H1; symmetry  in |- *; auto.
+Qed.   
+
+(* \paragraph{[O_CONSTANT_NEG]} *)
+
+Definition constant_neg (i : nat) (h : list proposition) :=
+  match nth_hyps i h with
+  | LeqTerm (Tint Z0) (Tint (Zneg n)) => absurd
+  | _ => h
+  end.
+
+Theorem constant_neg_valid : forall i : nat, valid_hyps (constant_neg i).
+
+unfold valid_hyps, constant_neg in |- *; intros;
+ generalize (nth_valid ep e i lp); Simplify; simpl in |- *;
+ unfold Zle in |- *; simpl in |- *; intros H1; elim H1;
+ [ assumption | trivial ].
+Qed.   
+
+(* \paragraph{[NOT_EXACT_DIVIDE]} *)
+Definition not_exact_divide (k1 k2 : Z) (body : term) 
+  (t i : nat) (l : list proposition) :=
+  match nth_hyps i l with
+  | EqTerm (Tint Z0) b =>
+      match
+        eq_term (scalar_norm_add t (Tplus (Tmult body (Tint k1)) (Tint k2)))
+          b
+      with
+      | true =>
+          match (k2 ?= 0)%Z with
+          | Datatypes.Gt =>
+              match (k1 ?= k2)%Z with
+              | Datatypes.Gt => absurd
+              | _ => l
+              end
+          | _ => l
+          end
+      | false => l
+      end
+  | _ => l
+  end.
+
+Theorem not_exact_divide_valid :
+ forall (k1 k2 : Z) (body : term) (t i : nat),
+ valid_hyps (not_exact_divide k1 k2 body t i).
+
+unfold valid_hyps, not_exact_divide in |- *; intros;
+ generalize (nth_valid ep e i lp); Simplify;
+ elim_eq_term (scalar_norm_add t (Tplus (Tmult body (Tint k1)) (Tint k2))) t1;
+ auto; Simplify; intro H2; elim H2; simpl in |- *;
+ elim (scalar_norm_add_stable t e); simpl in |- *; 
+ intro H4; absurd ((interp_term e body * k1 + k2)%Z = 0%Z);
+ [ apply OMEGA4; assumption | symmetry  in |- *; auto ].
+
+Qed.
+
+(* \paragraph{[O_CONTRADICTION]} *)
+
+Definition contradiction (t i j : nat) (l : list proposition) :=
+  match nth_hyps i l with
+  | LeqTerm (Tint Z0) b1 =>
+      match nth_hyps j l with
+      | LeqTerm (Tint Z0) b2 =>
+          match fusion_cancel t (Tplus b1 b2) with
+          | Tint k =>
+              match (0 ?= k)%Z with
+              | Datatypes.Gt => absurd
+              | _ => l
+              end
+          | _ => l
+          end
+      | _ => l
+      end
+  | _ => l
+  end.
+
+Theorem contradiction_valid :
+ forall t i j : nat, valid_hyps (contradiction t i j).
+
+unfold valid_hyps, contradiction in |- *; intros t i j ep e l H;
+ generalize (nth_valid _ _ i _ H); generalize (nth_valid _ _ j _ H);
+ case (nth_hyps i l); auto; intros t1 t2; case t1; 
+ auto; intros z; case z; auto; case (nth_hyps j l); 
+ auto; intros t3 t4; case t3; auto; intros z'; case z'; 
+ auto; simpl in |- *; intros H1 H2;
+ generalize (refl_equal (interp_term e (fusion_cancel t (Tplus t2 t4))));
+ pattern (fusion_cancel t (Tplus t2 t4)) at 2 3 in |- *;
+ case (fusion_cancel t (Tplus t2 t4)); simpl in |- *; 
+ auto; intro k; elim (fusion_cancel_stable t); simpl in |- *; 
+ intro E; generalize (OMEGA2 _ _ H2 H1); rewrite E; 
+ case k; auto; unfold Zle in |- *; simpl in |- *; intros p H3; 
+ elim H3; auto.
+
+Qed.
+
+(* \paragraph{[O_NEGATE_CONTRADICT]} *)
+
+Definition negate_contradict (i1 i2 : nat) (h : list proposition) :=
+  match nth_hyps i1 h with
+  | EqTerm (Tint Z0) b1 =>
+      match nth_hyps i2 h with
+      | NeqTerm (Tint Z0) b2 =>
+          match eq_term b1 b2 with
+          | true => absurd
+          | false => h
+          end
+      | _ => h
+      end
+  | NeqTerm (Tint Z0) b1 =>
+      match nth_hyps i2 h with
+      | EqTerm (Tint Z0) b2 =>
+          match eq_term b1 b2 with
+          | true => absurd
+          | false => h
+          end
+      | _ => h
+      end
+  | _ => h
+  end.
+
+Definition negate_contradict_inv (t i1 i2 : nat) (h : list proposition) :=
+  match nth_hyps i1 h with
+  | EqTerm (Tint Z0) b1 =>
+      match nth_hyps i2 h with
+      | NeqTerm (Tint Z0) b2 =>
+          match eq_term b1 (scalar_norm t (Tmult b2 (Tint (-1)))) with
+          | true => absurd
+          | false => h
+          end
+      | _ => h
+      end
+  | NeqTerm (Tint Z0) b1 =>
+      match nth_hyps i2 h with
+      | EqTerm (Tint Z0) b2 =>
+          match eq_term b1 (scalar_norm t (Tmult b2 (Tint (-1)))) with
+          | true => absurd
+          | false => h
+          end
+      | _ => h
+      end
+  | _ => h
+  end.
+
+Theorem negate_contradict_valid :
+ forall i j : nat, valid_hyps (negate_contradict i j).
+
+unfold valid_hyps, negate_contradict in |- *; intros i j ep e l H;
+ generalize (nth_valid _ _ i _ H); generalize (nth_valid _ _ j _ H);
+ case (nth_hyps i l); auto; intros t1 t2; case t1; 
+ auto; intros z; case z; auto; case (nth_hyps j l); 
+ auto; intros t3 t4; case t3; auto; intros z'; case z'; 
+ auto; simpl in |- *; intros H1 H2;
+ [ elim_eq_term t2 t4; intro H3;
+    [ elim H1; elim H3; assumption | assumption ]
+ | elim_eq_term t2 t4; intro H3;
+    [ elim H2; rewrite H3; assumption | assumption ] ].
+
+Qed.
+
+Theorem negate_contradict_inv_valid :
+ forall t i j : nat, valid_hyps (negate_contradict_inv t i j).
+
+
+unfold valid_hyps, negate_contradict_inv in |- *; intros t i j ep e l H;
+ generalize (nth_valid _ _ i _ H); generalize (nth_valid _ _ j _ H);
+ case (nth_hyps i l); auto; intros t1 t2; case t1; 
+ auto; intros z; case z; auto; case (nth_hyps j l); 
+ auto; intros t3 t4; case t3; auto; intros z'; case z'; 
+ auto; simpl in |- *; intros H1 H2;
+ (pattern (eq_term t2 (scalar_norm t (Tmult t4 (Tint (-1))))) in |- *;
+   apply bool_ind2; intro Aux;
+   [ generalize (eq_term_true t2 (scalar_norm t (Tmult t4 (Tint (-1)))) Aux);
+      clear Aux
+   | generalize (eq_term_false t2 (scalar_norm t (Tmult t4 (Tint (-1)))) Aux);
+      clear Aux ]);
+ [ intro H3; elim H1; generalize H2; rewrite H3;
+    rewrite <- (scalar_norm_stable t e); simpl in |- *;
+    elim (interp_term e t4); simpl in |- *; auto; intros p H4;
+    discriminate H4
+ | auto
+ | intro H3; elim H2; rewrite H3; elim (scalar_norm_stable t e);
+    simpl in |- *; elim H1; simpl in |- *; trivial
+ | auto ].
+
+Qed.
+
+(* \subsubsection{Tactiques générant une nouvelle équation} *)
+(* \paragraph{[O_SUM]}
+ C'est une oper2 valide mais elle traite plusieurs cas à la fois (suivant
+ les opérateurs de comparaison des deux arguments) d'où une
+ preuve un peu compliquée. On utilise quelques lemmes qui sont des 
+ généralisations des théorèmes utilisés par OMEGA. *)
+
+Definition sum (k1 k2 : Z) (trace : list t_fusion)
+  (prop1 prop2 : proposition) :=
+  match prop1 with
+  | EqTerm (Tint Z0) b1 =>
+      match prop2 with
+      | EqTerm (Tint Z0) b2 =>
+          EqTerm (Tint 0)
+            (fusion trace (Tplus (Tmult b1 (Tint k1)) (Tmult b2 (Tint k2))))
+      | LeqTerm (Tint Z0) b2 =>
+          match (k2 ?= 0)%Z with
+          | Datatypes.Gt =>
+              LeqTerm (Tint 0)
+                (fusion trace
+                   (Tplus (Tmult b1 (Tint k1)) (Tmult b2 (Tint k2))))
+          | _ => TrueTerm
+          end
+      | _ => TrueTerm
+      end
+  | LeqTerm (Tint Z0) b1 =>
+      match (k1 ?= 0)%Z with
+      | Datatypes.Gt =>
+          match prop2 with
+          | EqTerm (Tint Z0) b2 =>
+              LeqTerm (Tint 0)
+                (fusion trace
+                   (Tplus (Tmult b1 (Tint k1)) (Tmult b2 (Tint k2))))
+          | LeqTerm (Tint Z0) b2 =>
+              match (k2 ?= 0)%Z with
+              | Datatypes.Gt =>
+                  LeqTerm (Tint 0)
+                    (fusion trace
+                       (Tplus (Tmult b1 (Tint k1)) (Tmult b2 (Tint k2))))
+              | _ => TrueTerm
+              end
+          | _ => TrueTerm
+          end
+      | _ => TrueTerm
+      end
+  | NeqTerm (Tint Z0) b1 =>
+      match prop2 with
+      | EqTerm (Tint Z0) b2 =>
+          match eq_Z k1 0 with
+          | true => TrueTerm
+          | false =>
+              NeqTerm (Tint 0)
+                (fusion trace
+                   (Tplus (Tmult b1 (Tint k1)) (Tmult b2 (Tint k2))))
+          end
+      | _ => TrueTerm
+      end
+  | _ => TrueTerm
+  end.
+
+Theorem sum1 :
+ forall a b c d : Z, 0%Z = a -> 0%Z = b -> 0%Z = (a * c + b * d)%Z.
+
+intros; elim H; elim H0; simpl in |- *; auto.
+Qed.
+  
+Theorem sum2 :
+ forall a b c d : Z,
+ (0 <= d)%Z -> 0%Z = a -> (0 <= b)%Z -> (0 <= a * c + b * d)%Z.
+
+intros; elim H0; simpl in |- *; generalize H H1; case b; case d;
+ unfold Zle in |- *; simpl in |- *; auto.  
+Qed.
+
+Theorem sum3 :
+ forall a b c d : Z,
+ (0 <= c)%Z ->
+ (0 <= d)%Z -> (0 <= a)%Z -> (0 <= b)%Z -> (0 <= a * c + b * d)%Z.
+
+intros a b c d; case a; case b; case c; case d; unfold Zle in |- *;
+ simpl in |- *; auto. 
+Qed.
+
+Theorem sum4 : forall k : Z, (k ?= 0)%Z = Datatypes.Gt -> (0 <= k)%Z.
+
+intro; case k; unfold Zle in |- *; simpl in |- *; auto; intros; discriminate.  
+Qed.
+
+Theorem sum5 :
+ forall a b c d : Z,
+ c <> 0%Z -> 0%Z <> a -> 0%Z = b -> 0%Z <> (a * c + b * d)%Z.
+
+intros a b c d H1 H2 H3; elim H3; simpl in |- *; rewrite Zplus_comm;
+ simpl in |- *; generalize H1 H2; case a; case c; simpl in |- *; 
+ intros; try discriminate; assumption.
+Qed.
+
+
+Theorem sum_valid :
+ forall (k1 k2 : Z) (t : list t_fusion), valid2 (sum k1 k2 t).
+
+unfold valid2 in |- *; intros k1 k2 t ep e p1 p2; unfold sum in |- *;
+ Simplify; simpl in |- *; auto; try elim (fusion_stable t); 
+ simpl in |- *; intros;
+ [ apply sum1; assumption
+ | apply sum2; try assumption; apply sum4; assumption
+ | rewrite Zplus_comm; apply sum2; try assumption; apply sum4; assumption
+ | apply sum3; try assumption; apply sum4; assumption
+ | elim_eq_Z k1 0%Z; simpl in |- *; auto; elim (fusion_stable t);
+    simpl in |- *; intros; unfold Zne in |- *; apply sum5; 
+    assumption ].
+Qed.
+
+(* \paragraph{[O_EXACT_DIVIDE]}
+   c'est une oper1 valide mais on préfère une substitution a ce point la *)
+
+Definition exact_divide (k : Z) (body : term) (t : nat)
+  (prop : proposition) :=
+  match prop with
+  | EqTerm (Tint Z0) b =>
+      match eq_term (scalar_norm t (Tmult body (Tint k))) b with
+      | true =>
+          match eq_Z k 0 with
+          | true => TrueTerm
+          | false => EqTerm (Tint 0) body
+          end
+      | false => TrueTerm
+      end
+  | _ => TrueTerm
+  end.
+
+Theorem exact_divide_valid :
+ forall (k : Z) (t : term) (n : nat), valid1 (exact_divide k t n).
+
+
+unfold valid1, exact_divide in |- *; intros k1 k2 t ep e p1; Simplify;
+ simpl in |- *; auto; elim_eq_term (scalar_norm t (Tmult k2 (Tint k1))) t1;
+ simpl in |- *; auto; elim_eq_Z k1 0%Z; simpl in |- *; 
+ auto; intros H1 H2; elim H2; elim scalar_norm_stable; 
+ simpl in |- *; generalize H1; case (interp_term e k2); 
+ try trivial;
+ (case k1; simpl in |- *;
+   [ intros; absurd (0%Z = 0%Z); assumption
+   | intros p2 p3 H3 H4; discriminate H4
+   | intros p2 p3 H3 H4; discriminate H4 ]).
+
+Qed.
+
+
+
+(* \paragraph{[O_DIV_APPROX]}
+  La preuve reprend le schéma de la précédente mais on
+  est sur une opération de type valid1 et non sur une opération terminale. *)
+
+Definition divide_and_approx (k1 k2 : Z) (body : term) 
+  (t : nat) (prop : proposition) :=
+  match prop with
+  | LeqTerm (Tint Z0) b =>
+      match
+        eq_term (scalar_norm_add t (Tplus (Tmult body (Tint k1)) (Tint k2)))
+          b
+      with
+      | true =>
+          match (k1 ?= 0)%Z with
+          | Datatypes.Gt =>
+              match (k1 ?= k2)%Z with
+              | Datatypes.Gt => LeqTerm (Tint 0) body
+              | _ => prop
+              end
+          | _ => prop
+          end
+      | false => prop
+      end
+  | _ => prop
+  end.
+
+Theorem divide_and_approx_valid :
+ forall (k1 k2 : Z) (body : term) (t : nat),
+ valid1 (divide_and_approx k1 k2 body t).
+
+unfold valid1, divide_and_approx in |- *; intros k1 k2 body t ep e p1;
+ Simplify;
+ elim_eq_term (scalar_norm_add t (Tplus (Tmult body (Tint k1)) (Tint k2))) t1;
+ Simplify; auto; intro E; elim E; simpl in |- *;
+ elim (scalar_norm_add_stable t e); simpl in |- *; 
+ intro H1; apply Zmult_le_approx with (3 := H1); assumption.
+Qed.
+
+(* \paragraph{[MERGE_EQ]} *)
+
+Definition merge_eq (t : nat) (prop1 prop2 : proposition) :=
+  match prop1 with
+  | LeqTerm (Tint Z0) b1 =>
+      match prop2 with
+      | LeqTerm (Tint Z0) b2 =>
+          match eq_term b1 (scalar_norm t (Tmult b2 (Tint (-1)))) with
+          | true => EqTerm (Tint 0) b1
+          | false => TrueTerm
+          end
+      | _ => TrueTerm
+      end
+  | _ => TrueTerm
+  end.
+
+Theorem merge_eq_valid : forall n : nat, valid2 (merge_eq n).
+
+unfold valid2, merge_eq in |- *; intros n ep e p1 p2; Simplify; simpl in |- *;
+ auto; elim (scalar_norm_stable n e); simpl in |- *; 
+ intros; symmetry  in |- *; apply OMEGA8 with (2 := H0);
+ [ assumption | elim Zopp_eq_mult_neg_1; trivial ].
+Qed.
+
+
+
+(* \paragraph{[O_CONSTANT_NUL]} *)
+
+Definition constant_nul (i : nat) (h : list proposition) :=
+  match nth_hyps i h with
+  | NeqTerm (Tint Z0) (Tint Z0) => absurd
+  | _ => h
+  end.
+
+Theorem constant_nul_valid : forall i : nat, valid_hyps (constant_nul i).
+
+unfold valid_hyps, constant_nul in |- *; intros;
+ generalize (nth_valid ep e i lp); Simplify; simpl in |- *;
+ unfold Zne in |- *; intro H1; absurd (0%Z = 0%Z); 
+ auto.
+Qed.
+
+(* \paragraph{[O_STATE]} *)
+
+Definition state (m : Z) (s : step) (prop1 prop2 : proposition) :=
+  match prop1 with
+  | EqTerm (Tint Z0) b1 =>
+      match prop2 with
+      | EqTerm (Tint Z0) (Tplus b2 (Topp b3)) =>
+          EqTerm (Tint 0)
+            (rewrite s (Tplus b1 (Tmult (Tplus (Topp b3) b2) (Tint m))))
+      | _ => TrueTerm
+      end
+  | _ => TrueTerm
+  end.
+
+Theorem state_valid : forall (m : Z) (s : step), valid2 (state m s).
+
+unfold valid2 in |- *; intros m s ep e p1 p2; unfold state in |- *; Simplify;
+ simpl in |- *; auto; elim (rewrite_stable s e); simpl in |- *; 
+ intros H1 H2; elim H1;
+ rewrite (Zplus_comm (- interp_term e t5) (interp_term e t3)); 
+ elim H2; simpl in |- *; reflexivity.
+
+Qed.
+
+(* \subsubsection{Tactiques générant plusieurs but}
+   \paragraph{[O_SPLIT_INEQ]} 
+   La seule pour le moment (tant que la normalisation n'est pas réfléchie). *)
+
+Definition split_ineq (i t : nat)
+  (f1 f2 : list proposition -> list (list proposition))
+  (l : list proposition) :=
+  match nth_hyps i l with
+  | NeqTerm (Tint Z0) b1 =>
+      f1 (LeqTerm (Tint 0) (add_norm t (Tplus b1 (Tint (-1)))) :: l) ++
+      f2
+        (LeqTerm (Tint 0)
+           (scalar_norm_add t (Tplus (Tmult b1 (Tint (-1))) (Tint (-1))))
+         :: l)
+  | _ => l :: nil
+  end.
+
+Theorem split_ineq_valid :
+ forall (i t : nat) (f1 f2 : list proposition -> list (list proposition)),
+ valid_list_hyps f1 ->
+ valid_list_hyps f2 -> valid_list_hyps (split_ineq i t f1 f2).
+
+unfold valid_list_hyps, split_ineq in |- *; intros i t f1 f2 H1 H2 ep e lp H;
+ generalize (nth_valid _ _ i _ H); case (nth_hyps i lp); 
+ simpl in |- *; auto; intros t1 t2; case t1; simpl in |- *; 
+ auto; intros z; case z; simpl in |- *; auto; intro H3; 
+ apply append_valid; elim (OMEGA19 (interp_term e t2));
+ [ intro H4; left; apply H1; simpl in |- *; elim (add_norm_stable t);
+    simpl in |- *; auto
+ | intro H4; right; apply H2; simpl in |- *; elim (scalar_norm_add_stable t);
+    simpl in |- *; auto
+ | generalize H3; unfold Zne, not in |- *; intros E1 E2; apply E1;
+    symmetry  in |- *; trivial ].
+Qed.
+
+
+(* \subsection{La fonction de rejeu de la trace} *)
+
+Fixpoint execute_omega (t : t_omega) (l : list proposition) {struct t} :
+ list (list proposition) :=
+  match t with
+  | O_CONSTANT_NOT_NUL n =>
+      (fun a : list proposition => a :: nil) (constant_not_nul n l)
+  | O_CONSTANT_NEG n =>
+      (fun a : list proposition => a :: nil) (constant_neg n l)
+  | O_DIV_APPROX k1 k2 body t cont n =>
+      execute_omega cont (apply_oper_1 n (divide_and_approx k1 k2 body t) l)
+  | O_NOT_EXACT_DIVIDE k1 k2 body t i =>
+      (fun a : list proposition => a :: nil)
+        (not_exact_divide k1 k2 body t i l)
+  | O_EXACT_DIVIDE k body t cont n =>
+      execute_omega cont (apply_oper_1 n (exact_divide k body t) l)
+  | O_SUM k1 i1 k2 i2 t cont =>
+      execute_omega cont (apply_oper_2 i1 i2 (sum k1 k2 t) l)
+  | O_CONTRADICTION t i j =>
+      (fun a : list proposition => a :: nil) (contradiction t i j l)
+  | O_MERGE_EQ t i1 i2 cont =>
+      execute_omega cont (apply_oper_2 i1 i2 (merge_eq t) l)
+  | O_SPLIT_INEQ t i cont1 cont2 =>
+      split_ineq i t (execute_omega cont1) (execute_omega cont2) l
+  | O_CONSTANT_NUL i =>
+      (fun a : list proposition => a :: nil) (constant_nul i l)
+  | O_NEGATE_CONTRADICT i j =>
+      (fun a : list proposition => a :: nil) (negate_contradict i j l)
+  | O_NEGATE_CONTRADICT_INV t i j =>
+      (fun a : list proposition => a :: nil) (negate_contradict_inv t i j l)
+  | O_STATE m s i1 i2 cont =>
+      execute_omega cont (apply_oper_2 i1 i2 (state m s) l)
+  end.
+
+Theorem omega_valid : forall t : t_omega, valid_list_hyps (execute_omega t).
+
+simple induction t; simpl in |- *;
+ [ unfold valid_list_hyps in |- *; simpl in |- *; intros; left;
+    apply (constant_not_nul_valid n ep e lp H)
+ | unfold valid_list_hyps in |- *; simpl in |- *; intros; left;
+    apply (constant_neg_valid n ep e lp H)
+ | unfold valid_list_hyps, valid_hyps in |- *;
+    intros k1 k2 body n t' Ht' m ep e lp H; apply Ht';
+    apply
+     (apply_oper_1_valid m (divide_and_approx k1 k2 body n)
+        (divide_and_approx_valid k1 k2 body n) ep e lp H)
+ | unfold valid_list_hyps in |- *; simpl in |- *; intros; left;
+    apply (not_exact_divide_valid z z0 t0 n n0 ep e lp H)
+ | unfold valid_list_hyps, valid_hyps in |- *;
+    intros k body n t' Ht' m ep e lp H; apply Ht';
+    apply
+     (apply_oper_1_valid m (exact_divide k body n)
+        (exact_divide_valid k body n) ep e lp H)
+ | unfold valid_list_hyps, valid_hyps in |- *;
+    intros k1 i1 k2 i2 trace t' Ht' ep e lp H; apply Ht';
+    apply
+     (apply_oper_2_valid i1 i2 (sum k1 k2 trace) (sum_valid k1 k2 trace) ep e
+        lp H)
+ | unfold valid_list_hyps in |- *; simpl in |- *; intros; left;
+    apply (contradiction_valid n n0 n1 ep e lp H)
+ | unfold valid_list_hyps, valid_hyps in |- *;
+    intros trace i1 i2 t' Ht' ep e lp H; apply Ht';
+    apply
+     (apply_oper_2_valid i1 i2 (merge_eq trace) (merge_eq_valid trace) ep e
+        lp H)
+ | intros t' i k1 H1 k2 H2; unfold valid_list_hyps in |- *; simpl in |- *;
+    intros ep e lp H;
+    apply
+     (split_ineq_valid i t' (execute_omega k1) (execute_omega k2) H1 H2 ep e
+        lp H)
+ | unfold valid_list_hyps in |- *; simpl in |- *; intros i ep e lp H; left;
+    apply (constant_nul_valid i ep e lp H)
+ | unfold valid_list_hyps in |- *; simpl in |- *; intros i j ep e lp H; left;
+    apply (negate_contradict_valid i j ep e lp H)
+ | unfold valid_list_hyps in |- *; simpl in |- *; intros n i j ep e lp H;
+    left; apply (negate_contradict_inv_valid n i j ep e lp H)
+ | unfold valid_list_hyps, valid_hyps in |- *;
+    intros m s i1 i2 t' Ht' ep e lp H; apply Ht';
+    apply (apply_oper_2_valid i1 i2 (state m s) (state_valid m s) ep e lp H) ].
+Qed.
+
+
+(* \subsection{Les opérations globales sur le but} 
+   \subsubsection{Normalisation} *)
+
+Definition move_right (s : step) (p : proposition) :=
+  match p with
+  | EqTerm t1 t2 => EqTerm (Tint 0) (rewrite s (Tplus t1 (Topp t2)))
+  | LeqTerm t1 t2 => LeqTerm (Tint 0) (rewrite s (Tplus t2 (Topp t1)))
+  | GeqTerm t1 t2 => LeqTerm (Tint 0) (rewrite s (Tplus t1 (Topp t2)))
+  | LtTerm t1 t2 =>
+      LeqTerm (Tint 0) (rewrite s (Tplus (Tplus t2 (Tint (-1))) (Topp t1)))
+  | GtTerm t1 t2 =>
+      LeqTerm (Tint 0) (rewrite s (Tplus (Tplus t1 (Tint (-1))) (Topp t2)))
+  | NeqTerm t1 t2 => NeqTerm (Tint 0) (rewrite s (Tplus t1 (Topp t2)))
+  | p => p
+  end.
+
+Theorem Zne_left_2 : forall x y : Z, Zne x y -> Zne 0 (x + - y).
+unfold Zne, not in |- *; intros x y H1 H2; apply H1;
+ apply (Zplus_reg_l (- y)); rewrite Zplus_comm; elim H2; 
+ rewrite Zplus_opp_l; trivial.
+Qed.
+
+Theorem move_right_valid : forall s : step, valid1 (move_right s).
+
+unfold valid1, move_right in |- *; intros s ep e p; Simplify; simpl in |- *;
+ elim (rewrite_stable s e); simpl in |- *;
+ [ symmetry  in |- *; apply Zegal_left; assumption
+ | intro; apply Zle_left; assumption
+ | intro; apply Zge_left; assumption
+ | intro; apply Zgt_left; assumption
+ | intro; apply Zlt_left; assumption
+ | intro; apply Zne_left_2; assumption ].
+Qed.
+
+Definition do_normalize (i : nat) (s : step) := apply_oper_1 i (move_right s).
+
+Theorem do_normalize_valid :
+ forall (i : nat) (s : step), valid_hyps (do_normalize i s).
+
+intros; unfold do_normalize in |- *; apply apply_oper_1_valid;
+ apply move_right_valid.
+Qed.
+
+Fixpoint do_normalize_list (l : list step) (i : nat) 
+ (h : list proposition) {struct l} : list proposition :=
+  match l with
+  | s :: l' => do_normalize_list l' (S i) (do_normalize i s h)
+  | nil => h
+  end.
+
+Theorem do_normalize_list_valid :
+ forall (l : list step) (i : nat), valid_hyps (do_normalize_list l i).
+
+simple induction l; simpl in |- *; unfold valid_hyps in |- *;
+ [ auto
+ | intros a l' Hl' i ep e lp H; unfold valid_hyps in Hl'; apply Hl';
+    apply (do_normalize_valid i a ep e lp); assumption ].
+Qed.
+
+Theorem normalize_goal :
+ forall (s : list step) (ep : PropList) (env : list Z) (l : list proposition),
+ (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+  interp_goal_concl envp env FalseTerm l) ep env (do_normalize_list s 0 l) ->
+ (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+  interp_goal_concl envp env FalseTerm l) ep env l.
+
+intros; apply valid_goal with (2 := H); apply do_normalize_list_valid.
+Qed.
+
+(* \subsubsection{Exécution de la trace} *)
+
+Theorem execute_goal :
+ forall (t : t_omega) (ep : PropList) (env : list Z) (l : list proposition),
+ interp_list_goal ep env (execute_omega t l) ->
+ (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+  interp_goal_concl envp env FalseTerm l) ep env l.
+
+intros; apply (goal_valid (execute_omega t) (omega_valid t) ep env l H).
+Qed.
+
+
+Theorem append_goal :
+ forall (ep : PropList) (e : list Z) (l1 l2 : list (list proposition)),
+ interp_list_goal ep e l1 /\ interp_list_goal ep e l2 ->
+ interp_list_goal ep e (l1 ++ l2).
+
+intros ep e; simple induction l1;
+ [ simpl in |- *; intros l2 (H1, H2); assumption
+ | simpl in |- *; intros h1 t1 HR l2 ((H1, H2), H3); split; auto ].
+
+Qed.
+
+Require Import Decidable.
+
+(* A simple decidability checker : if the proposition belongs to the
+   simple grammar describe below then it is decidable. Proof is by 
+   induction and uses well known theorem about arithmetic and propositional
+   calculus *)
+
+Fixpoint decidability (p : proposition) : bool :=
+  match p with
+  | EqTerm _ _ => true
+  | LeqTerm _ _ => true
+  | GeqTerm _ _ => true
+  | GtTerm _ _ => true
+  | LtTerm _ _ => true
+  | NeqTerm _ _ => true
+  | FalseTerm => true
+  | TrueTerm => true
+  | Tnot t => decidability t
+  | Tand t1 t2 => decidability t1 && decidability t2
+  | Timp t1 t2 => decidability t1 && decidability t2
+  | Tor t1 t2 => decidability t1 && decidability t2
+  | Tprop _ => false
+  end.
+
+Theorem decidable_correct :
+ forall (ep : PropList) (e : list Z) (p : proposition),
+ decidability p = true -> decidable (interp_proposition ep e p).
+
+simple induction p; simpl in |- *; intros;
+ [ apply dec_eq
+ | apply dec_Zle
+ | left; auto
+ | right; unfold not in |- *; auto
+ | apply dec_not; auto
+ | apply dec_Zge
+ | apply dec_Zgt
+ | apply dec_Zlt
+ | apply dec_Zne
+ | apply dec_or; elim andb_prop with (1 := H1); auto
+ | apply dec_and; elim andb_prop with (1 := H1); auto
+ | apply dec_imp; elim andb_prop with (1 := H1); auto
+ | discriminate H ].
+
+Qed.
+
+(* An interpretation function for a complete goal with an explicit
+   conclusion. We use an intermediate fixpoint. *)
+
+Fixpoint interp_full_goal (envp : PropList) (env : list Z) 
+ (c : proposition) (l : list proposition) {struct l} : Prop :=
+  match l with
+  | nil => interp_proposition envp env c
+  | p' :: l' =>
+      interp_proposition envp env p' -> interp_full_goal envp env c l'
+  end.
+
+Definition interp_full (ep : PropList) (e : list Z)
+  (lc : list proposition * proposition) : Prop :=
+  match lc with
+  | (l, c) => interp_full_goal ep e c l
+  end.
+
+(* Relates the interpretation of a complete goal with the interpretation
+   of its hypothesis and conclusion *)
+
+Theorem interp_full_false :
+ forall (ep : PropList) (e : list Z) (l : list proposition) (c : proposition),
+ (interp_hyps ep e l -> interp_proposition ep e c) -> interp_full ep e (l, c).
+
+simple induction l; unfold interp_full in |- *; simpl in |- *;
+ [ auto | intros a l1 H1 c H2 H3; apply H1; auto ].
+
+Qed.
+
+(* Push the conclusion in the list of hypothesis using a double negation
+   If the decidability cannot be "proven", then just forget about the 
+   conclusion (equivalent of replacing it with false) *)
+
+Definition to_contradict (lc : list proposition * proposition) :=
+  match lc with
+  | (l, c) => if decidability c then Tnot c :: l else l
+  end.
+
+(* The previous operation is valid in the sense that the new list of
+   hypothesis implies the original goal *)
+
+Theorem to_contradict_valid :
+ forall (ep : PropList) (e : list Z) (lc : list proposition * proposition),
+ (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+  interp_goal_concl envp env FalseTerm l) ep e (to_contradict lc) ->
+ interp_full ep e lc.
+
+intros ep e lc; case lc; intros l c; simpl in |- *;
+ pattern (decidability c) in |- *; apply bool_ind2;
+ [ simpl in |- *; intros H H1; apply interp_full_false; intros H2;
+    apply not_not;
+    [ apply decidable_correct; assumption
+    | unfold not at 1 in |- *; intro H3; apply hyps_to_goal with (2 := H2);
+       auto ]
+ | intros H1 H2; apply interp_full_false; intro H3;
+    elim hyps_to_goal with (1 := H2); assumption ].
+Qed.
+
+(* [map_cons x l] adds [x] at the head of each list in [l] (which is a list
+   of lists *)
+
+Fixpoint map_cons (A : Set) (x : A) (l : list (list A)) {struct l} :
+ list (list A) :=
+  match l with
+  | nil => nil
+  | l :: ll => (x :: l) :: map_cons A x ll
+  end.
+
+(* This function breaks up a list of hypothesis in a list of simpler 
+   list of hypothesis that together implie the original one. The goal
+   of all this is to transform the goal in a list of solvable problems. 
+   Note that :
+	- we need a way to drive the analysis as some hypotheis may not
+          require a split. 
+        - this procedure must be perfectly mimicked by the ML part otherwise
+          hypothesis will get desynchronised and this will be a mess.
+ *)
+ 
+Fixpoint destructure_hyps (nn : nat) (ll : list proposition) {struct nn} :
+ list (list proposition) :=
+  match nn with
+  | O => ll :: nil
+  | S n =>
+      match ll with
+      | nil => nil :: nil
+      | Tor p1 p2 :: l =>
+          destructure_hyps n (p1 :: l) ++ destructure_hyps n (p2 :: l)
+      | Tand p1 p2 :: l => destructure_hyps n (p1 :: p2 :: l)
+      | Timp p1 p2 :: l =>
+          if decidability p1
+          then
+           destructure_hyps n (Tnot p1 :: l) ++ destructure_hyps n (p2 :: l)
+          else map_cons _ (Timp p1 p2) (destructure_hyps n l)
+      | Tnot p :: l =>
+          match p with
+          | Tnot p1 =>
+              if decidability p1
+              then destructure_hyps n (p1 :: l)
+              else map_cons _ (Tnot (Tnot p1)) (destructure_hyps n l)
+          | Tor p1 p2 => destructure_hyps n (Tnot p1 :: Tnot p2 :: l)
+          | Tand p1 p2 =>
+              if decidability p1
+              then
+               destructure_hyps n (Tnot p1 :: l) ++
+               destructure_hyps n (Tnot p2 :: l)
+              else map_cons _ (Tnot p) (destructure_hyps n l)
+          | _ => map_cons _ (Tnot p) (destructure_hyps n l)
+          end
+      | x :: l => map_cons _ x (destructure_hyps n l)
+      end
+  end.
+
+Theorem map_cons_val :
+ forall (ep : PropList) (e : list Z) (p : proposition)
+   (l : list (list proposition)),
+ interp_proposition ep e p ->
+ interp_list_hyps ep e l -> interp_list_hyps ep e (map_cons _ p l).
+
+simple induction l; simpl in |- *; [ auto | intros; elim H1; intro H2; auto ].
+Qed.
+
+Hint Resolve map_cons_val append_valid decidable_correct.
+
+Theorem destructure_hyps_valid :
+ forall n : nat, valid_list_hyps (destructure_hyps n).
+
+simple induction n;
+ [ unfold valid_list_hyps in |- *; simpl in |- *; auto
+ | unfold valid_list_hyps at 2 in |- *; intros n1 H ep e lp; case lp;
+    [ simpl in |- *; auto
+    | intros p l; case p;
+       try
+        (simpl in |- *; intros; apply map_cons_val; simpl in |- *; elim H0;
+          auto);
+       [ intro p'; case p';
+          try
+           (simpl in |- *; intros; apply map_cons_val; simpl in |- *; elim H0;
+             auto);
+          [ simpl in |- *; intros p1 (H1, H2);
+             pattern (decidability p1) in |- *; apply bool_ind2; 
+             intro H3;
+             [ apply H; simpl in |- *; split;
+                [ apply not_not; auto | assumption ]
+             | auto ]
+          | simpl in |- *; intros p1 p2 (H1, H2); apply H; simpl in |- *;
+             elim not_or with (1 := H1); auto
+          | simpl in |- *; intros p1 p2 (H1, H2);
+             pattern (decidability p1) in |- *; apply bool_ind2; 
+             intro H3;
+             [ apply append_valid; elim not_and with (2 := H1);
+                [ intro; left; apply H; simpl in |- *; auto
+                | intro; right; apply H; simpl in |- *; auto
+                | auto ]
+             | auto ] ]
+       | simpl in |- *; intros p1 p2 (H1, H2); apply append_valid;
+          (elim H1; intro H3; simpl in |- *; [ left | right ]); 
+          apply H; simpl in |- *; auto
+       | simpl in |- *; intros; apply H; simpl in |- *; tauto
+       | simpl in |- *; intros p1 p2 (H1, H2);
+          pattern (decidability p1) in |- *; apply bool_ind2; 
+          intro H3;
+          [ apply append_valid; elim imp_simp with (2 := H1);
+             [ intro H4; left; simpl in |- *; apply H; simpl in |- *; auto
+             | intro H4; right; simpl in |- *; apply H; simpl in |- *; auto
+             | auto ]
+          | auto ] ] ] ].
+
+Qed.
+
+Definition prop_stable (f : proposition -> proposition) :=
+  forall (ep : PropList) (e : list Z) (p : proposition),
+  interp_proposition ep e p <-> interp_proposition ep e (f p).
+
+Definition p_apply_left (f : proposition -> proposition) 
+  (p : proposition) :=
+  match p with
+  | Timp x y => Timp (f x) y
+  | Tor x y => Tor (f x) y
+  | Tand x y => Tand (f x) y
+  | Tnot x => Tnot (f x)
+  | x => x
+  end.
+
+Theorem p_apply_left_stable :
+ forall f : proposition -> proposition,
+ prop_stable f -> prop_stable (p_apply_left f).
+
+unfold prop_stable in |- *; intros f H ep e p; split;
+ (case p; simpl in |- *; auto; intros p1; elim (H ep e p1); tauto).
+Qed.
+
+Definition p_apply_right (f : proposition -> proposition)
+  (p : proposition) :=
+  match p with
+  | Timp x y => Timp x (f y)
+  | Tor x y => Tor x (f y)
+  | Tand x y => Tand x (f y)
+  | Tnot x => Tnot (f x)
+  | x => x
+  end.
+
+Theorem p_apply_right_stable :
+ forall f : proposition -> proposition,
+ prop_stable f -> prop_stable (p_apply_right f).
+
+unfold prop_stable in |- *; intros f H ep e p; split;
+ (case p; simpl in |- *; auto;
+   [ intros p1; elim (H ep e p1); tauto
+   | intros p1 p2; elim (H ep e p2); tauto
+   | intros p1 p2; elim (H ep e p2); tauto
+   | intros p1 p2; elim (H ep e p2); tauto ]).
+Qed.
+
+Definition p_invert (f : proposition -> proposition) 
+  (p : proposition) :=
+  match p with
+  | EqTerm x y => Tnot (f (NeqTerm x y))
+  | LeqTerm x y => Tnot (f (GtTerm x y))
+  | GeqTerm x y => Tnot (f (LtTerm x y))
+  | GtTerm x y => Tnot (f (LeqTerm x y))
+  | LtTerm x y => Tnot (f (GeqTerm x y))
+  | NeqTerm x y => Tnot (f (EqTerm x y))
+  | x => x
+  end.
+
+Theorem p_invert_stable :
+ forall f : proposition -> proposition,
+ prop_stable f -> prop_stable (p_invert f).
+
+unfold prop_stable in |- *; intros f H ep e p; split;
+ (case p; simpl in |- *; auto;
+   [ intros t1 t2; elim (H ep e (NeqTerm t1 t2)); simpl in |- *;
+      unfold Zne in |- *;
+      generalize (dec_eq (interp_term e t1) (interp_term e t2));
+      unfold decidable in |- *; tauto
+   | intros t1 t2; elim (H ep e (GtTerm t1 t2)); simpl in |- *;
+      unfold Zgt in |- *;
+      generalize (dec_Zgt (interp_term e t1) (interp_term e t2));
+      unfold decidable, Zgt, Zle in |- *; tauto
+   | intros t1 t2; elim (H ep e (LtTerm t1 t2)); simpl in |- *;
+      unfold Zlt in |- *;
+      generalize (dec_Zlt (interp_term e t1) (interp_term e t2));
+      unfold decidable, Zge in |- *; tauto
+   | intros t1 t2; elim (H ep e (LeqTerm t1 t2)); simpl in |- *;
+      generalize (dec_Zgt (interp_term e t1) (interp_term e t2));
+      unfold Zle, Zgt in |- *; unfold decidable in |- *; 
+      tauto
+   | intros t1 t2; elim (H ep e (GeqTerm t1 t2)); simpl in |- *;
+      generalize (dec_Zlt (interp_term e t1) (interp_term e t2));
+      unfold Zge, Zlt in |- *; unfold decidable in |- *; 
+      tauto
+   | intros t1 t2; elim (H ep e (EqTerm t1 t2)); simpl in |- *;
+      generalize (dec_eq (interp_term e t1) (interp_term e t2));
+      unfold decidable, Zne in |- *; tauto ]).
+Qed.
+
+Theorem Zlt_left_inv : forall x y : Z, (0 <= y + -1 + - x)%Z -> (x < y)%Z.
+
+intros; apply Zsucc_lt_reg; apply Zle_lt_succ;
+ apply (fun a b : Z => Zplus_le_reg_r a b (-1 + - x)); 
+ rewrite Zplus_assoc; unfold Zsucc in |- *; rewrite (Zplus_assoc_reverse x);
+ rewrite (Zplus_assoc y); simpl in |- *; rewrite Zplus_0_r;
+ rewrite Zplus_opp_r; assumption.
+Qed.
+
+Theorem move_right_stable : forall s : step, prop_stable (move_right s).
+
+unfold move_right, prop_stable in |- *; intros s ep e p; split;
+ [ Simplify; simpl in |- *; elim (rewrite_stable s e); simpl in |- *;
+    [ symmetry  in |- *; apply Zegal_left; assumption
+    | intro; apply Zle_left; assumption
+    | intro; apply Zge_left; assumption
+    | intro; apply Zgt_left; assumption
+    | intro; apply Zlt_left; assumption
+    | intro; apply Zne_left_2; assumption ]
+ | case p; simpl in |- *; intros; auto; generalize H; elim (rewrite_stable s);
+    simpl in |- *; intro H1;
+    [ rewrite (Zplus_0_r_reverse (interp_term e t0)); rewrite H1;
+       rewrite Zplus_permute; rewrite Zplus_opp_r; 
+       rewrite Zplus_0_r; trivial
+    | apply (fun a b : Z => Zplus_le_reg_r a b (- interp_term e t));
+       rewrite Zplus_opp_r; assumption
+    | apply Zle_ge;
+       apply (fun a b : Z => Zplus_le_reg_r a b (- interp_term e t0));
+       rewrite Zplus_opp_r; assumption
+    | apply Zlt_gt; apply Zlt_left_inv; assumption
+    | apply Zlt_left_inv; assumption
+    | unfold Zne, not in |- *; unfold Zne in H1; intro H2; apply H1;
+       rewrite H2; rewrite Zplus_opp_r; trivial ] ].
+Qed.
+
+
+Fixpoint p_rewrite (s : p_step) : proposition -> proposition :=
+  match s with
+  | P_LEFT s => p_apply_left (p_rewrite s)
+  | P_RIGHT s => p_apply_right (p_rewrite s)
+  | P_STEP s => move_right s
+  | P_INVERT s => p_invert (move_right s)
+  | P_NOP => fun p : proposition => p
+  end.
+
+Theorem p_rewrite_stable : forall s : p_step, prop_stable (p_rewrite s).
+
+
+simple induction s; simpl in |- *;
+ [ intros; apply p_apply_left_stable; trivial
+ | intros; apply p_apply_right_stable; trivial
+ | intros; apply p_invert_stable; apply move_right_stable
+ | apply move_right_stable
+ | unfold prop_stable in |- *; simpl in |- *; intros; split; auto ].
+Qed.
+
+Fixpoint normalize_hyps (l : list h_step) (lh : list proposition) {struct l}
+ : list proposition :=
+  match l with
+  | nil => lh
+  | pair_step i s :: r => normalize_hyps r (apply_oper_1 i (p_rewrite s) lh)
+  end.
+
+Theorem normalize_hyps_valid :
+ forall l : list h_step, valid_hyps (normalize_hyps l).
+
+simple induction l; unfold valid_hyps in |- *; simpl in |- *;
+ [ auto
+ | intros n_s r; case n_s; intros n s H ep e lp H1; apply H;
+    apply apply_oper_1_valid;
+    [ unfold valid1 in |- *; intros ep1 e1 p1 H2;
+       elim (p_rewrite_stable s ep1 e1 p1); auto
+    | assumption ] ].
+Qed.
+
+Theorem normalize_hyps_goal :
+ forall (s : list h_step) (ep : PropList) (env : list Z)
+   (l : list proposition),
+ (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+  interp_goal_concl envp env FalseTerm l) ep env (normalize_hyps s l) ->
+ (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+  interp_goal_concl envp env FalseTerm l) ep env l.
+
+intros; apply valid_goal with (2 := H); apply normalize_hyps_valid.
+Qed.
+
+Fixpoint extract_hyp_pos (s : list direction) (p : proposition) {struct s} :
+ proposition :=
+  match s with
+  | D_left :: l =>
+      match p with
+      | Tand x y => extract_hyp_pos l x
+      | _ => p
+      end
+  | D_right :: l =>
+      match p with
+      | Tand x y => extract_hyp_pos l y
+      | _ => p
+      end
+  | D_mono :: l => match p with
+                   | Tnot x => extract_hyp_neg l x
+                   | _ => p
+                   end
+  | _ => p
+  end
+ 
+ with extract_hyp_neg (s : list direction) (p : proposition) {struct s} :
+ proposition :=
+  match s with
+  | D_left :: l =>
+      match p with
+      | Tor x y => extract_hyp_neg l x
+      | Timp x y => if decidability x then extract_hyp_pos l x else Tnot p
+      | _ => Tnot p
+      end
+  | D_right :: l =>
+      match p with
+      | Tor x y => extract_hyp_neg l y
+      | Timp x y => extract_hyp_neg l y
+      | _ => Tnot p
+      end
+  | D_mono :: l =>
+      match p with
+      | Tnot x => if decidability x then extract_hyp_pos l x else Tnot p
+      | _ => Tnot p
+      end
+  | _ =>
+      match p with
+      | Tnot x => if decidability x then x else Tnot p
+      | _ => Tnot p
+      end
+  end.
+
+Definition co_valid1 (f : proposition -> proposition) :=
+  forall (ep : PropList) (e : list Z) (p1 : proposition),
+  interp_proposition ep e (Tnot p1) -> interp_proposition ep e (f p1).
+
+Theorem extract_valid :
+ forall s : list direction,
+ valid1 (extract_hyp_pos s) /\ co_valid1 (extract_hyp_neg s).
+
+unfold valid1, co_valid1 in |- *; simple induction s;
+ [ split;
+    [ simpl in |- *; auto
+    | intros ep e p1; case p1; simpl in |- *; auto; intro p;
+       pattern (decidability p) in |- *; apply bool_ind2;
+       [ intro H; generalize (decidable_correct ep e p H);
+          unfold decidable in |- *; tauto
+       | simpl in |- *; auto ] ]
+ | intros a s' (H1, H2); simpl in H2; split; intros ep e p; case a; auto;
+    case p; auto; simpl in |- *; intros;
+    (apply H1; tauto) ||
+      (apply H2; tauto) ||
+        (pattern (decidability p0) in |- *; apply bool_ind2;
+          [ intro H3; generalize (decidable_correct ep e p0 H3);
+             unfold decidable in |- *; intro H4; apply H1; 
+             tauto
+          | intro; tauto ]) ].
+
+Qed.
+
+Fixpoint decompose_solve (s : e_step) (h : list proposition) {struct s} :
+ list (list proposition) :=
+  match s with
+  | E_SPLIT i dl s1 s2 =>
+      match extract_hyp_pos dl (nth_hyps i h) with
+      | Tor x y => decompose_solve s1 (x :: h) ++ decompose_solve s2 (y :: h)
+      | Tnot (Tand x y) =>
+          if decidability x
+          then
+           decompose_solve s1 (Tnot x :: h) ++
+           decompose_solve s2 (Tnot y :: h)
+          else h :: nil
+      | _ => h :: nil
+      end
+  | E_EXTRACT i dl s1 =>
+      decompose_solve s1 (extract_hyp_pos dl (nth_hyps i h) :: h)
+  | E_SOLVE t => execute_omega t h
+  end.
+
+Theorem decompose_solve_valid :
+ forall s : e_step, valid_list_goal (decompose_solve s).
+
+intro s; apply goal_valid; unfold valid_list_hyps in |- *; elim s;
+ simpl in |- *; intros;
+ [ cut (interp_proposition ep e1 (extract_hyp_pos l (nth_hyps n lp)));
+    [ case (extract_hyp_pos l (nth_hyps n lp)); simpl in |- *; auto;
+       [ intro p; case p; simpl in |- *; auto; intros p1 p2 H2;
+          pattern (decidability p1) in |- *; apply bool_ind2;
+          [ intro H3; generalize (decidable_correct ep e1 p1 H3); intro H4;
+             apply append_valid; elim H4; intro H5;
+             [ right; apply H0; simpl in |- *; tauto
+             | left; apply H; simpl in |- *; tauto ]
+          | simpl in |- *; auto ]
+       | intros p1 p2 H2; apply append_valid; simpl in |- *; elim H2;
+          [ intros H3; left; apply H; simpl in |- *; auto
+          | intros H3; right; apply H0; simpl in |- *; auto ] ]
+    | elim (extract_valid l); intros H2 H3; apply H2; apply nth_valid; auto ]
+ | intros; apply H; simpl in |- *; split;
+    [ elim (extract_valid l); intros H2 H3; apply H2; apply nth_valid; auto
+    | auto ]
+ | apply omega_valid with (1 := H) ].
+
+Qed.
+
+(* \subsection{La dernière étape qui élimine tous les séquents inutiles} *)
+
+Definition valid_lhyps
+  (f : list (list proposition) -> list (list proposition)) :=
+  forall (ep : PropList) (e : list Z) (lp : list (list proposition)),
+  interp_list_hyps ep e lp -> interp_list_hyps ep e (f lp).
+
+Fixpoint reduce_lhyps (lp : list (list proposition)) :
+ list (list proposition) :=
+  match lp with
+  | (FalseTerm :: nil) :: lp' => reduce_lhyps lp'
+  | x :: lp' => x :: reduce_lhyps lp'
+  | nil => nil (A:=list proposition)
+  end.
+
+Theorem reduce_lhyps_valid : valid_lhyps reduce_lhyps.
+
+unfold valid_lhyps in |- *; intros ep e lp; elim lp;
+ [ simpl in |- *; auto
+ | intros a l HR; elim a;
+    [ simpl in |- *; tauto
+    | intros a1 l1; case l1; case a1; simpl in |- *; try tauto ] ].
+Qed.
+
+Theorem do_reduce_lhyps :
+ forall (envp : PropList) (env : list Z) (l : list (list proposition)),
+ interp_list_goal envp env (reduce_lhyps l) -> interp_list_goal envp env l.
+
+intros envp env l H; apply list_goal_to_hyps; intro H1;
+ apply list_hyps_to_goal with (1 := H); apply reduce_lhyps_valid; 
+ assumption.
+Qed.
+
+Definition concl_to_hyp (p : proposition) :=
+  if decidability p then Tnot p else TrueTerm.
+
+Definition do_concl_to_hyp :
+  forall (envp : PropList) (env : list Z) (c : proposition)
+    (l : list proposition),
+  (fun (envp : PropList) (env : list Z) (l : list proposition) =>
+   interp_goal_concl envp env FalseTerm l) envp env (
+    concl_to_hyp c :: l) -> interp_goal_concl envp env c l.
+
+simpl in |- *; intros envp env c l; induction  l as [| a l Hrecl];
+ [ simpl in |- *; unfold concl_to_hyp in |- *;
+    pattern (decidability c) in |- *; apply bool_ind2;
+    [ intro H; generalize (decidable_correct envp env c H);
+       unfold decidable in |- *; simpl in |- *; tauto
+    | simpl in |- *; intros H1 H2; elim H2; trivial ]
+ | simpl in |- *; tauto ].
+Qed.
+
+Definition omega_tactic (t1 : e_step) (t2 : list h_step) 
+  (c : proposition) (l : list proposition) :=
+  reduce_lhyps (decompose_solve t1 (normalize_hyps t2 (concl_to_hyp c :: l))).
+
+Theorem do_omega :
+ forall (t1 : e_step) (t2 : list h_step) (envp : PropList) 
+   (env : list Z) (c : proposition) (l : list proposition),
+ interp_list_goal envp env (omega_tactic t1 t2 c l) ->
+ interp_goal_concl envp env c l.
+
+unfold omega_tactic in |- *; intros; apply do_concl_to_hyp;
+ apply (normalize_hyps_goal t2); apply (decompose_solve_valid t1);
+ apply do_reduce_lhyps; assumption.
+Qed.
\ No newline at end of file
diff --git a/contrib/romega/const_omega.ml b/contrib/romega/const_omega.ml
new file mode 100644
index 00000000..3b2a7d31
--- /dev/null
+++ b/contrib/romega/const_omega.ml
@@ -0,0 +1,488 @@
+(*************************************************************************
+
+   PROJET RNRT Calife - 2001
+   Author: Pierre Crégut - France Télécom R&D
+   Licence : LGPL version 2.1
+
+ *************************************************************************)
+
+let module_refl_name = "ReflOmegaCore"
+let module_refl_path = ["Coq"; "romega"; module_refl_name]
+
+type result = 
+   Kvar of string
+ | Kapp of string * Term.constr list
+ | Kimp of Term.constr * Term.constr
+ | Kufo;;
+
+let destructurate t =
+  let c, args = Term.decompose_app t in
+  let env = Global.env() in
+  match Term.kind_of_term c, args with
+    | Term.Const sp, args ->
+	Kapp (Names.string_of_id
+		(Nametab.id_of_global (Libnames.ConstRef sp)),
+	      args)
+    | Term.Construct csp , args ->
+	Kapp (Names.string_of_id
+		(Nametab.id_of_global (Libnames.ConstructRef csp)),
+	      args)
+    | Term.Ind isp, args ->
+	Kapp (Names.string_of_id
+		(Nametab.id_of_global (Libnames.IndRef isp)),
+	      args)
+    | Term.Var id,[] -> Kvar(Names.string_of_id id)
+    | Term.Prod (Names.Anonymous,typ,body), [] -> Kimp(typ,body)
+    | Term.Prod (Names.Name _,_,_),[] ->
+	Util.error "Omega: Not a quantifier-free goal"
+    | _ -> Kufo
+
+exception Destruct
+
+let dest_const_apply t = 
+  let f,args = Term.decompose_app t in 
+  let ref = 
+  match Term.kind_of_term f with 
+    | Term.Const sp         -> Libnames.ConstRef sp
+    | Term.Construct csp -> Libnames.ConstructRef csp
+    | Term.Ind isp       -> Libnames.IndRef isp
+    | _ -> raise Destruct
+  in  Nametab.id_of_global ref, args
+
+let recognize_number t =
+  let rec loop t =
+    let f,l = dest_const_apply t in
+    match Names.string_of_id f,l with
+       "xI",[t] -> 1 + 2 * loop t
+     | "xO",[t] -> 2 * loop t 
+     | "xH",[] -> 1
+     | _ -> failwith "not a number" in
+  let f,l = dest_const_apply t in
+    match Names.string_of_id f,l with
+       "Zpos",[t] -> loop t | "Zneg",[t] -> - (loop t) | "Z0",[] -> 0
+     | _ -> failwith "not a number";;
+
+
+let logic_dir = ["Coq";"Logic";"Decidable"]
+
+let coq_modules =
+  Coqlib.init_modules @ [logic_dir] @ Coqlib.arith_modules @ Coqlib.zarith_base_modules
+    @ [["Coq"; "omega"; "OmegaLemmas"]]
+    @ [["Coq"; "Lists"; (if !Options.v7 then "PolyList" else "List")]]
+    @ [module_refl_path]
+
+
+let constant = Coqlib.gen_constant_in_modules "Omega" coq_modules
+
+let coq_xH = lazy (constant "xH")
+let coq_xO = lazy (constant "xO")
+let coq_xI = lazy (constant "xI")
+let coq_ZERO = lazy (constant "Z0")
+let coq_POS = lazy (constant "Zpos")
+let coq_NEG = lazy (constant "Zneg")
+let coq_Z = lazy (constant "Z")
+let coq_relation = lazy (constant  "comparison")
+let coq_SUPERIEUR = lazy (constant "SUPERIEUR")
+let coq_INFEEIEUR = lazy (constant "INFERIEUR")
+let coq_EGAL = lazy (constant "EGAL")
+let coq_Zplus = lazy (constant "Zplus")
+let coq_Zmult = lazy (constant  "Zmult")
+let coq_Zopp = lazy (constant "Zopp")
+
+let coq_Zminus = lazy (constant "Zminus")
+let coq_Zs = lazy (constant  "Zs")
+let coq_Zgt = lazy (constant "Zgt")
+let coq_Zle = lazy (constant "Zle")
+let coq_inject_nat = lazy (constant  "inject_nat")
+
+(* Peano *)
+let coq_le = lazy(constant "le")
+let coq_gt = lazy(constant "gt")
+
+(* Integers *)
+let coq_nat = lazy(constant "nat")
+let coq_S = lazy(constant "S")
+let coq_O = lazy(constant "O")
+let coq_minus = lazy(constant "minus")
+
+(* Logic *)
+let coq_eq = lazy(constant  "eq")
+let coq_refl_equal = lazy(constant  "refl_equal")
+let coq_and = lazy(constant "and")
+let coq_not = lazy(constant "not")
+let coq_or = lazy(constant "or")
+let coq_true = lazy(constant "true")
+let coq_false = lazy(constant "false")
+let coq_ex = lazy(constant "ex")
+let coq_I = lazy(constant "I")
+
+(* Lists *)
+let coq_cons =  lazy (constant "cons")
+let coq_nil =  lazy (constant "nil")
+
+let coq_pcons = lazy (constant "Pcons")
+let coq_pnil = lazy (constant "Pnil")
+
+let coq_h_step = lazy (constant "h_step")
+let coq_pair_step = lazy (constant  "pair_step")
+let coq_p_left = lazy (constant  "P_LEFT")
+let coq_p_right = lazy (constant  "P_RIGHT")
+let coq_p_invert = lazy (constant  "P_INVERT")
+let coq_p_step = lazy (constant  "P_STEP")
+let coq_p_nop = lazy (constant  "P_NOP")
+
+
+let coq_t_int = lazy (constant  "Tint")
+let coq_t_plus = lazy (constant  "Tplus")
+let coq_t_mult = lazy (constant  "Tmult")
+let coq_t_opp = lazy (constant  "Topp")
+let coq_t_minus = lazy (constant  "Tminus")
+let coq_t_var = lazy (constant  "Tvar")
+
+let coq_p_eq = lazy (constant  "EqTerm")
+let coq_p_leq = lazy (constant  "LeqTerm")
+let coq_p_geq = lazy (constant  "GeqTerm")
+let coq_p_lt = lazy (constant  "LtTerm")
+let coq_p_gt = lazy (constant  "GtTerm")
+let coq_p_neq = lazy (constant  "NeqTerm")
+let coq_p_true = lazy (constant  "TrueTerm")
+let coq_p_false = lazy (constant  "FalseTerm")
+let coq_p_not = lazy (constant  "Tnot")
+let coq_p_or = lazy (constant  "Tor")
+let coq_p_and = lazy (constant  "Tand")
+let coq_p_imp = lazy (constant  "Timp")
+let coq_p_prop = lazy (constant  "Tprop")
+
+let coq_proposition = lazy (constant  "proposition")
+let coq_interp_sequent = lazy (constant  "interp_goal_concl")
+let coq_normalize_sequent = lazy (constant  "normalize_goal")
+let coq_execute_sequent = lazy (constant  "execute_goal")
+let coq_do_concl_to_hyp =  lazy (constant  "do_concl_to_hyp")
+let coq_sequent_to_hyps = lazy (constant  "goal_to_hyps")
+let coq_normalize_hyps_goal =
+  lazy (constant  "normalize_hyps_goal")
+
+(* Constructors for shuffle tactic *)
+let coq_t_fusion =  lazy (constant  "t_fusion")
+let coq_f_equal =  lazy (constant  "F_equal")
+let coq_f_cancel =  lazy (constant  "F_cancel")
+let coq_f_left =  lazy (constant  "F_left")
+let coq_f_right =  lazy (constant  "F_right")
+
+(* Constructors for reordering tactics *)
+let coq_step = lazy (constant  "step")
+let coq_c_do_both = lazy (constant  "C_DO_BOTH")
+let coq_c_do_left = lazy (constant  "C_LEFT")
+let coq_c_do_right = lazy (constant  "C_RIGHT")
+let coq_c_do_seq = lazy (constant  "C_SEQ")
+let coq_c_nop = lazy (constant  "C_NOP")
+let coq_c_opp_plus = lazy (constant  "C_OPP_PLUS")
+let coq_c_opp_opp = lazy (constant  "C_OPP_OPP")
+let coq_c_opp_mult_r = lazy (constant  "C_OPP_MULT_R")
+let coq_c_opp_one = lazy (constant  "C_OPP_ONE")
+let coq_c_reduce = lazy (constant  "C_REDUCE")
+let coq_c_mult_plus_distr = lazy (constant  "C_MULT_PLUS_DISTR")
+let coq_c_opp_left = lazy (constant  "C_MULT_OPP_LEFT")
+let coq_c_mult_assoc_r = lazy (constant  "C_MULT_ASSOC_R")
+let coq_c_plus_assoc_r = lazy (constant  "C_PLUS_ASSOC_R")
+let coq_c_plus_assoc_l = lazy (constant  "C_PLUS_ASSOC_L")
+let coq_c_plus_permute = lazy (constant  "C_PLUS_PERMUTE")
+let coq_c_plus_sym = lazy (constant  "C_PLUS_SYM")
+let coq_c_red0 = lazy (constant  "C_RED0")
+let coq_c_red1 = lazy (constant  "C_RED1")
+let coq_c_red2 = lazy (constant  "C_RED2")
+let coq_c_red3 = lazy (constant  "C_RED3")
+let coq_c_red4 = lazy (constant  "C_RED4")
+let coq_c_red5 = lazy (constant  "C_RED5")
+let coq_c_red6 = lazy (constant  "C_RED6")
+let coq_c_mult_opp_left = lazy (constant  "C_MULT_OPP_LEFT")
+let coq_c_mult_assoc_reduced = 
+  lazy (constant  "C_MULT_ASSOC_REDUCED")
+let coq_c_minus = lazy (constant  "C_MINUS")
+let coq_c_mult_sym = lazy (constant  "C_MULT_SYM")
+
+let coq_s_constant_not_nul = lazy (constant  "O_CONSTANT_NOT_NUL")
+let coq_s_constant_neg = lazy (constant  "O_CONSTANT_NEG")
+let coq_s_div_approx = lazy (constant  "O_DIV_APPROX")
+let coq_s_not_exact_divide = lazy (constant  "O_NOT_EXACT_DIVIDE")
+let coq_s_exact_divide = lazy (constant  "O_EXACT_DIVIDE")
+let coq_s_sum = lazy (constant  "O_SUM")
+let coq_s_state = lazy (constant  "O_STATE")
+let coq_s_contradiction = lazy (constant  "O_CONTRADICTION")
+let coq_s_merge_eq = lazy (constant  "O_MERGE_EQ")
+let coq_s_split_ineq =lazy (constant  "O_SPLIT_INEQ")
+let coq_s_constant_nul =lazy (constant  "O_CONSTANT_NUL")
+let coq_s_negate_contradict =lazy (constant  "O_NEGATE_CONTRADICT")
+let coq_s_negate_contradict_inv =lazy (constant  "O_NEGATE_CONTRADICT_INV")
+
+(* construction for the [extract_hyp] tactic *)
+let coq_direction = lazy  (constant  "direction")
+let coq_d_left = lazy  (constant  "D_left")
+let coq_d_right = lazy  (constant  "D_right")
+let coq_d_mono = lazy  (constant  "D_mono")
+
+let coq_e_split = lazy  (constant  "E_SPLIT")
+let coq_e_extract = lazy  (constant  "E_EXTRACT")
+let coq_e_solve = lazy  (constant  "E_SOLVE")
+
+let coq_decompose_solve_valid = 
+  lazy (constant   "decompose_solve_valid")
+let coq_do_reduce_lhyps = lazy (constant   "do_reduce_lhyps")
+let coq_do_omega = lazy (constant  "do_omega")
+
+(**
+let constant dir s =
+   try
+     Libnames.constr_of_reference 
+       (Nametab.absolute_reference
+	  (Libnames.make_path
+             (Names.make_dirpath (List.map Names.id_of_string (List.rev dir)))
+             (Names.id_of_string s)))
+   with e -> print_endline (String.concat "." dir); print_endline s;
+             raise e
+
+let path_fast_integer = ["Coq"; "ZArith"; "fast_integer"]
+let path_zarith_aux = ["Coq"; "ZArith"; "zarith_aux"]
+let path_logic = ["Coq"; "Init";"Logic"]
+let path_datatypes = ["Coq"; "Init";"Datatypes"]
+let path_peano = ["Coq"; "Init"; "Peano"]
+let path_list = ["Coq"; "Lists"; "PolyList"]
+
+let coq_xH = lazy (constant path_fast_integer "xH")
+let coq_xO = lazy (constant path_fast_integer "xO")
+let coq_xI = lazy (constant path_fast_integer "xI")
+let coq_ZERO = lazy (constant path_fast_integer "ZERO")
+let coq_POS = lazy (constant path_fast_integer "POS")
+let coq_NEG = lazy (constant path_fast_integer "NEG")
+let coq_Z = lazy (constant path_fast_integer "Z")
+let coq_relation = lazy (constant path_fast_integer "relation")
+let coq_SUPERIEUR = lazy (constant path_fast_integer "SUPERIEUR")
+let coq_INFEEIEUR = lazy (constant path_fast_integer "INFERIEUR")
+let coq_EGAL = lazy (constant path_fast_integer "EGAL")
+let coq_Zplus = lazy (constant path_fast_integer "Zplus")
+let coq_Zmult = lazy (constant path_fast_integer "Zmult")
+let coq_Zopp = lazy (constant path_fast_integer "Zopp")
+
+(* auxiliaires zarith *)
+let coq_Zminus = lazy (constant path_zarith_aux "Zminus")
+let coq_Zs = lazy (constant path_zarith_aux "Zs")
+let coq_Zgt = lazy (constant path_zarith_aux "Zgt")
+let coq_Zle = lazy (constant path_zarith_aux "Zle")
+let coq_inject_nat = lazy (constant path_zarith_aux "inject_nat")
+
+(* Peano *)
+let coq_le = lazy(constant path_peano "le")
+let coq_gt = lazy(constant path_peano "gt")
+
+(* Integers *)
+let coq_nat = lazy(constant path_datatypes "nat")
+let coq_S = lazy(constant path_datatypes "S")
+let coq_O = lazy(constant path_datatypes "O")
+
+(* Minus *)
+let coq_minus = lazy(constant ["Arith"; "Minus"] "minus")
+
+(* Logic *)
+let coq_eq = lazy(constant path_logic "eq")
+let coq_refl_equal = lazy(constant path_logic "refl_equal")
+let coq_and = lazy(constant path_logic "and")
+let coq_not = lazy(constant path_logic "not")
+let coq_or = lazy(constant path_logic "or")
+let coq_true = lazy(constant path_logic "true")
+let coq_false = lazy(constant path_logic "false")
+let coq_ex = lazy(constant path_logic "ex")
+let coq_I = lazy(constant path_logic "I")
+
+(* Lists *)
+let coq_cons =  lazy (constant path_list  "cons")
+let coq_nil =  lazy (constant path_list "nil")
+
+let coq_pcons = lazy (constant module_refl_path "Pcons")
+let coq_pnil = lazy (constant module_refl_path "Pnil")
+
+let coq_h_step = lazy (constant module_refl_path "h_step")
+let coq_pair_step = lazy (constant module_refl_path "pair_step")
+let coq_p_left = lazy (constant module_refl_path "P_LEFT")
+let coq_p_right = lazy (constant module_refl_path "P_RIGHT")
+let coq_p_invert = lazy (constant module_refl_path "P_INVERT")
+let coq_p_step = lazy (constant module_refl_path "P_STEP")
+let coq_p_nop = lazy (constant module_refl_path "P_NOP")
+
+
+let coq_t_int = lazy (constant module_refl_path "Tint")
+let coq_t_plus = lazy (constant module_refl_path "Tplus")
+let coq_t_mult = lazy (constant module_refl_path "Tmult")
+let coq_t_opp = lazy (constant module_refl_path "Topp")
+let coq_t_minus = lazy (constant module_refl_path "Tminus")
+let coq_t_var = lazy (constant module_refl_path "Tvar")
+
+let coq_p_eq = lazy (constant module_refl_path "EqTerm")
+let coq_p_leq = lazy (constant module_refl_path "LeqTerm")
+let coq_p_geq = lazy (constant module_refl_path "GeqTerm")
+let coq_p_lt = lazy (constant module_refl_path "LtTerm")
+let coq_p_gt = lazy (constant module_refl_path "GtTerm")
+let coq_p_neq = lazy (constant module_refl_path "NeqTerm")
+let coq_p_true = lazy (constant module_refl_path "TrueTerm")
+let coq_p_false = lazy (constant module_refl_path "FalseTerm")
+let coq_p_not = lazy (constant module_refl_path "Tnot")
+let coq_p_or = lazy (constant module_refl_path "Tor")
+let coq_p_and = lazy (constant module_refl_path "Tand")
+let coq_p_imp = lazy (constant module_refl_path "Timp")
+let coq_p_prop = lazy (constant module_refl_path "Tprop")
+
+let coq_proposition = lazy (constant module_refl_path "proposition")
+let coq_interp_sequent = lazy (constant module_refl_path "interp_goal_concl")
+let coq_normalize_sequent = lazy (constant module_refl_path "normalize_goal")
+let coq_execute_sequent = lazy (constant module_refl_path "execute_goal")
+let coq_do_concl_to_hyp =  lazy (constant module_refl_path "do_concl_to_hyp")
+let coq_sequent_to_hyps = lazy (constant module_refl_path "goal_to_hyps")
+let coq_normalize_hyps_goal =
+  lazy (constant module_refl_path "normalize_hyps_goal")
+
+(* Constructors for shuffle tactic *)
+let coq_t_fusion =  lazy (constant module_refl_path "t_fusion")
+let coq_f_equal =  lazy (constant module_refl_path "F_equal")
+let coq_f_cancel =  lazy (constant module_refl_path "F_cancel")
+let coq_f_left =  lazy (constant module_refl_path "F_left")
+let coq_f_right =  lazy (constant module_refl_path "F_right")
+
+(* Constructors for reordering tactics *)
+let coq_step = lazy (constant module_refl_path "step")
+let coq_c_do_both = lazy (constant module_refl_path "C_DO_BOTH")
+let coq_c_do_left = lazy (constant module_refl_path "C_LEFT")
+let coq_c_do_right = lazy (constant module_refl_path "C_RIGHT")
+let coq_c_do_seq = lazy (constant module_refl_path "C_SEQ")
+let coq_c_nop = lazy (constant module_refl_path "C_NOP")
+let coq_c_opp_plus = lazy (constant module_refl_path "C_OPP_PLUS")
+let coq_c_opp_opp = lazy (constant module_refl_path "C_OPP_OPP")
+let coq_c_opp_mult_r = lazy (constant module_refl_path "C_OPP_MULT_R")
+let coq_c_opp_one = lazy (constant module_refl_path "C_OPP_ONE")
+let coq_c_reduce = lazy (constant module_refl_path "C_REDUCE")
+let coq_c_mult_plus_distr = lazy (constant module_refl_path "C_MULT_PLUS_DISTR")
+let coq_c_opp_left = lazy (constant module_refl_path "C_MULT_OPP_LEFT")
+let coq_c_mult_assoc_r = lazy (constant module_refl_path "C_MULT_ASSOC_R")
+let coq_c_plus_assoc_r = lazy (constant module_refl_path "C_PLUS_ASSOC_R")
+let coq_c_plus_assoc_l = lazy (constant module_refl_path "C_PLUS_ASSOC_L")
+let coq_c_plus_permute = lazy (constant module_refl_path "C_PLUS_PERMUTE")
+let coq_c_plus_sym = lazy (constant module_refl_path "C_PLUS_SYM")
+let coq_c_red0 = lazy (constant module_refl_path "C_RED0")
+let coq_c_red1 = lazy (constant module_refl_path "C_RED1")
+let coq_c_red2 = lazy (constant module_refl_path "C_RED2")
+let coq_c_red3 = lazy (constant module_refl_path "C_RED3")
+let coq_c_red4 = lazy (constant module_refl_path "C_RED4")
+let coq_c_red5 = lazy (constant module_refl_path "C_RED5")
+let coq_c_red6 = lazy (constant module_refl_path "C_RED6")
+let coq_c_mult_opp_left = lazy (constant module_refl_path "C_MULT_OPP_LEFT")
+let coq_c_mult_assoc_reduced = 
+  lazy (constant module_refl_path "C_MULT_ASSOC_REDUCED")
+let coq_c_minus = lazy (constant module_refl_path "C_MINUS")
+let coq_c_mult_sym = lazy (constant module_refl_path "C_MULT_SYM")
+
+let coq_s_constant_not_nul = lazy (constant module_refl_path "O_CONSTANT_NOT_NUL")
+let coq_s_constant_neg = lazy (constant module_refl_path "O_CONSTANT_NEG")
+let coq_s_div_approx = lazy (constant module_refl_path "O_DIV_APPROX")
+let coq_s_not_exact_divide = lazy (constant module_refl_path "O_NOT_EXACT_DIVIDE")
+let coq_s_exact_divide = lazy (constant module_refl_path "O_EXACT_DIVIDE")
+let coq_s_sum = lazy (constant module_refl_path "O_SUM")
+let coq_s_state = lazy (constant module_refl_path "O_STATE")
+let coq_s_contradiction = lazy (constant module_refl_path "O_CONTRADICTION")
+let coq_s_merge_eq = lazy (constant module_refl_path "O_MERGE_EQ")
+let coq_s_split_ineq =lazy (constant module_refl_path "O_SPLIT_INEQ")
+let coq_s_constant_nul =lazy (constant module_refl_path "O_CONSTANT_NUL")
+let coq_s_negate_contradict =lazy (constant module_refl_path "O_NEGATE_CONTRADICT")
+let coq_s_negate_contradict_inv =lazy (constant module_refl_path "O_NEGATE_CONTRADICT_INV")
+
+(* construction for the [extract_hyp] tactic *)
+let coq_direction = lazy  (constant module_refl_path "direction")
+let coq_d_left = lazy  (constant module_refl_path "D_left")
+let coq_d_right = lazy  (constant module_refl_path "D_right")
+let coq_d_mono = lazy  (constant module_refl_path "D_mono")
+
+let coq_e_split = lazy  (constant module_refl_path "E_SPLIT")
+let coq_e_extract = lazy  (constant module_refl_path "E_EXTRACT")
+let coq_e_solve = lazy  (constant module_refl_path "E_SOLVE")
+
+let coq_decompose_solve_valid = 
+  lazy (constant  module_refl_path "decompose_solve_valid")
+let coq_do_reduce_lhyps = lazy (constant  module_refl_path "do_reduce_lhyps")
+let coq_do_omega = lazy (constant module_refl_path "do_omega")
+
+*)
+(* \subsection{Construction d'expressions} *)
+
+
+let mk_var v = Term.mkVar (Names.id_of_string v)
+let mk_plus t1 t2 = Term.mkApp (Lazy.force coq_Zplus,[|  t1; t2 |])
+let mk_times t1 t2 = Term.mkApp (Lazy.force coq_Zmult, [| t1; t2 |])
+let mk_minus t1 t2 = Term.mkApp (Lazy.force coq_Zminus, [| t1;t2 |])
+let mk_eq t1 t2 = Term.mkApp (Lazy.force coq_eq, [| Lazy.force coq_Z; t1; t2 |])
+let mk_le t1 t2 = Term.mkApp (Lazy.force coq_Zle, [|t1; t2 |])
+let mk_gt t1 t2 = Term.mkApp (Lazy.force coq_Zgt, [|t1; t2 |])
+let mk_inv t = Term.mkApp (Lazy.force coq_Zopp, [|t |])
+let mk_and t1 t2 =  Term.mkApp (Lazy.force coq_and, [|t1; t2 |])
+let mk_or t1 t2 =  Term.mkApp (Lazy.force coq_or, [|t1; t2 |])
+let mk_not t = Term.mkApp (Lazy.force coq_not, [|t |])
+let mk_eq_rel t1 t2 = Term.mkApp (Lazy.force coq_eq, [|
+				Lazy.force coq_relation; t1; t2 |])
+let mk_inj t = Term.mkApp (Lazy.force coq_inject_nat, [|t |])
+
+
+let do_left t = 
+  if t = Lazy.force coq_c_nop then Lazy.force coq_c_nop
+  else Term.mkApp (Lazy.force coq_c_do_left, [|t |] )
+
+let do_right t = 
+  if t = Lazy.force coq_c_nop then Lazy.force coq_c_nop
+  else Term.mkApp (Lazy.force coq_c_do_right, [|t |])
+
+let do_both t1 t2 = 
+  if t1 = Lazy.force coq_c_nop then do_right t2
+  else if t2 = Lazy.force coq_c_nop then do_left t1
+  else Term.mkApp (Lazy.force coq_c_do_both , [|t1; t2 |])
+
+let do_seq t1 t2 =
+  if t1 = Lazy.force coq_c_nop then t2
+  else if t2 = Lazy.force coq_c_nop then t1
+  else Term.mkApp (Lazy.force coq_c_do_seq, [|t1; t2 |])
+  
+let rec do_list = function
+  | [] -> Lazy.force coq_c_nop
+  | [x] -> x
+  | (x::l) -> do_seq x (do_list l)
+
+				 
+let mk_integer n =
+  let rec loop n = 
+    if n=1 then Lazy.force coq_xH else 
+      Term.mkApp ((if n mod 2 = 0 then Lazy.force coq_xO else Lazy.force coq_xI),
+		 [| loop (n/2) |]) in
+    
+    if n = 0 then Lazy.force coq_ZERO 
+    else Term.mkApp ((if n > 0 then Lazy.force coq_POS else Lazy.force coq_NEG),
+		     [| loop (abs n) |])
+
+let mk_Z = mk_integer
+
+let rec mk_nat = function
+  | 0 -> Lazy.force coq_O
+  | n -> Term.mkApp (Lazy.force coq_S, [| mk_nat (n-1) |])
+
+let mk_list typ l =
+  let rec loop = function
+    | [] ->
+	Term.mkApp (Lazy.force coq_nil, [|typ|])
+    | (step :: l) -> 
+	Term.mkApp (Lazy.force coq_cons, [|typ; step; loop l |]) in
+  loop l
+
+let mk_plist l =
+  let rec loop = function
+    | [] ->
+	(Lazy.force coq_pnil)
+    | (step :: l) -> 
+	Term.mkApp (Lazy.force coq_pcons, [| step; loop l |]) in
+  loop l
+
+
+let mk_shuffle_list l = mk_list (Lazy.force coq_t_fusion) l
+
diff --git a/contrib/romega/g_romega.ml4 b/contrib/romega/g_romega.ml4
new file mode 100644
index 00000000..386f7f28
--- /dev/null
+++ b/contrib/romega/g_romega.ml4
@@ -0,0 +1,15 @@
+(*************************************************************************
+
+   PROJET RNRT Calife - 2001
+   Author: Pierre Crégut - France Télécom R&D
+   Licence : LGPL version 2.1
+
+ *************************************************************************)
+
+(*i camlp4deps: "parsing/grammar.cma" i*)
+
+open Refl_omega
+
+TACTIC EXTEND ROmega
+  [ "ROmega" ] -> [ total_reflexive_omega_tactic ]
+END
diff --git a/contrib/romega/omega2.ml b/contrib/romega/omega2.ml
new file mode 100644
index 00000000..91aefc60
--- /dev/null
+++ b/contrib/romega/omega2.ml
@@ -0,0 +1,675 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+(**************************************************************************)
+(*                                                                        *)
+(* Omega: a solver of quantifier-free problems in Presburger Arithmetic   *)
+(*                                                                        *)
+(* Pierre Crégut (CNET, Lannion, France)                                  *)
+(*                                                                        *)
+(* 13/10/2002 : modified to cope with an external numbering of equations  *)
+(*   and hypothesis. Its use for Omega is not more complex and it makes   *)
+(*   things much simpler for the reflexive version where we should limit  *)
+(*   the number of source of numbering.                                   *)
+(**************************************************************************)
+
+open Names
+
+let flat_map f =
+ let rec flat_map_f = function
+   | [] -> [] 
+   | x :: l -> f x @ flat_map_f l
+ in 
+ flat_map_f
+
+let pp i = print_int i; print_newline (); flush stdout
+
+let debug = ref false
+
+let filter = List.partition
+
+let push v l = l := v :: !l
+
+let rec pgcd x y = if y = 0 then x else pgcd y (x mod y)
+
+let pgcd_l = function
+  | [] -> failwith "pgcd_l"
+  | x :: l -> List.fold_left pgcd x l
+
+let floor_div a b =
+  match a >=0 , b > 0 with
+    | true,true -> a / b
+    | false,false -> a / b
+    | true, false -> (a-1) / b - 1
+    | false,true  -> (a+1) / b - 1
+
+type coeff = {c: int ; v: int}
+
+type linear = coeff list
+
+type eqn_kind = EQUA | INEQ | DISE
+
+type afine = { 
+  (* a number uniquely identifying the equation *)
+  id: int ; 
+  (* a boolean true for an eq, false for an ineq (Sigma a_i x_i >= 0) *)
+  kind: eqn_kind; 
+  (* the variables and their coefficient *)
+  body: coeff list; 
+  (* a constant *)
+  constant: int }
+
+type state_action = {
+  st_new_eq : afine;
+  st_def    :  afine;
+  st_orig   : afine;
+  st_coef   : int;
+  st_var    : int }
+
+type action =
+  | DIVIDE_AND_APPROX of afine * afine * int * int
+  | NOT_EXACT_DIVIDE of afine * int
+  | FORGET_C of int
+  | EXACT_DIVIDE of afine * int
+  | SUM of int * (int * afine) * (int * afine)
+  | STATE of state_action 
+  | HYP of afine
+  | FORGET   of int * int
+  | FORGET_I of int * int
+  | CONTRADICTION of afine * afine
+  | NEGATE_CONTRADICT of afine * afine * bool
+  | MERGE_EQ of int * afine * int 
+  | CONSTANT_NOT_NUL of int * int
+  | CONSTANT_NUL of int
+  | CONSTANT_NEG of int * int
+  | SPLIT_INEQ of afine * (int * action list) * (int * action list)
+  | WEAKEN of int * int
+
+exception UNSOLVABLE
+
+exception NO_CONTRADICTION
+
+let display_eq print_var (l,e) =
+  let _ = 
+    List.fold_left 
+      (fun not_first f ->
+	 print_string 
+	   (if f.c < 0 then "- " else if not_first then "+ " else "");
+	 let c = abs f.c in
+	 if c = 1 then 
+	   Printf.printf "%s " (print_var f.v)
+	 else 
+	   Printf.printf "%d %s " c (print_var f.v); 
+	 true)
+      false l
+  in
+  if e > 0 then 
+    Printf.printf "+ %d " e 
+  else if e < 0 then 
+    Printf.printf "- %d " (abs e)
+
+let rec trace_length l =
+  let action_length accu = function
+    | SPLIT_INEQ (_,(_,l1),(_,l2)) ->
+	accu + 1 + trace_length l1 + trace_length l2
+    | _ -> accu + 1 in
+  List.fold_left action_length 0 l
+
+let operator_of_eq = function 
+  | EQUA -> "=" | DISE -> "!=" | INEQ -> ">="
+
+let kind_of = function
+  | EQUA -> "equation" | DISE -> "disequation" | INEQ -> "inequation"
+
+let display_system print_var l = 
+  List.iter 
+    (fun { kind=b; body=e; constant=c; id=id} -> 
+       print_int id; print_string ": ";
+       display_eq print_var (e,c); print_string (operator_of_eq b);
+       print_string "0\n") 
+    l;
+  print_string "------------------------\n\n"
+
+let display_inequations print_var l = 
+  List.iter (fun e -> display_eq print_var e;print_string ">= 0\n") l;
+  print_string "------------------------\n\n"
+
+let rec display_action print_var = function
+  | act :: l -> begin match act with
+      | DIVIDE_AND_APPROX (e1,e2,k,d) ->
+          Printf.printf 
+            "Inequation E%d is divided by %d and the constant coefficient is \
+            rounded by substracting %d.\n" e1.id k d
+      | NOT_EXACT_DIVIDE (e,k) ->
+          Printf.printf
+            "Constant in equation E%d is not divisible by the pgcd \
+            %d of its other coefficients.\n" e.id k
+      | EXACT_DIVIDE (e,k) ->
+          Printf.printf
+            "Equation E%d is divided by the pgcd \
+            %d of its coefficients.\n" e.id k
+      | WEAKEN (e,k) ->
+          Printf.printf 
+            "To ensure a solution in the dark shadow \
+            the equation E%d is weakened by %d.\n" e k
+      | SUM (e,(c1,e1),(c2,e2)) -> 
+          Printf.printf
+            "We state %s E%d = %d %s E%d + %d %s E%d.\n" 
+            (kind_of e1.kind) e c1 (kind_of e1.kind) e1.id c2 
+            (kind_of e2.kind) e2.id
+      | STATE { st_new_eq = e; st_coef = x} ->
+          Printf.printf "We define a new equation %d :" e.id; 
+          display_eq print_var (e.body,e.constant); 
+          print_string (operator_of_eq e.kind); print_string " 0\n"
+      | HYP e ->   
+          Printf.printf "We define %d :" e.id; 
+          display_eq print_var (e.body,e.constant); 
+          print_string (operator_of_eq e.kind); print_string " 0\n"
+      | FORGET_C e ->  Printf.printf "E%d is trivially satisfiable.\n" e
+      | FORGET (e1,e2) -> Printf.printf "E%d subsumes E%d.\n" e1 e2
+      | FORGET_I (e1,e2) -> Printf.printf "E%d subsumes E%d.\n" e1 e2
+      | MERGE_EQ (e,e1,e2) ->
+         Printf.printf "E%d and E%d can be merged into E%d.\n" e1.id e2 e
+      | CONTRADICTION (e1,e2) -> 
+          Printf.printf
+            "equations E%d and E%d implie a contradiction on their \
+            constant factors.\n" e1.id e2.id
+      | NEGATE_CONTRADICT(e1,e2,b) ->
+          Printf.printf
+            "Eqations E%d and E%d state that their body is at the same time
+            equal and different\n" e1.id e2.id
+      | CONSTANT_NOT_NUL (e,k) -> 
+          Printf.printf "equation E%d states %d=0.\n" e k
+      | CONSTANT_NEG(e,k) -> 
+          Printf.printf "equation E%d states %d >= 0.\n" e k
+      | CONSTANT_NUL e ->
+          Printf.printf "inequation E%d states 0 != 0.\n" e
+      | SPLIT_INEQ (e,(e1,l1),(e2,l2)) -> 
+          Printf.printf "equation E%d is split in E%d and E%d\n\n" e.id e1 e2;
+          display_action print_var l1;
+          print_newline ();
+          display_action print_var l2;
+          print_newline ()
+    end; display_action print_var l
+  | [] -> 
+      flush stdout
+
+(*""*)
+let default_print_var v = Printf.sprintf "XX%d" v
+
+let add_event, history, clear_history =
+  let accu = ref [] in
+  (fun (v:action) -> if !debug then display_action default_print_var [v]; push v accu),
+  (fun () -> !accu),
+  (fun () -> accu := [])
+
+let nf_linear = Sort.list (fun x y -> x.v > y.v)
+
+let nf ((b : bool),(e,(x : int))) = (b,(nf_linear e,x))
+
+let map_eq_linear f = 
+  let rec loop = function
+    | x :: l -> let c = f x.c in if c=0 then loop l else {v=x.v; c=c} :: loop l
+    | [] -> [] 
+  in
+  loop
+
+let map_eq_afine f e = 
+  { id = e.id; kind = e.kind; body = map_eq_linear f e.body; 
+    constant = f e.constant }
+
+let negate_eq = map_eq_afine (fun x -> -x)
+
+let rec sum p0 p1 = match (p0,p1) with 
+  | ([], l) -> l | (l, []) -> l
+  | (((x1::l1) as l1'), ((x2::l2) as l2')) -> 
+      if x1.v = x2.v then
+	let c = x1.c + x2.c in
+	if c = 0 then sum l1 l2 else {v=x1.v;c=c} :: sum l1 l2
+      else if x1.v > x2.v then 
+	x1 :: sum l1 l2'
+      else 
+	x2 :: sum l1' l2
+
+let sum_afine new_eq_id eq1 eq2 = 
+  { kind = eq1.kind; id = new_eq_id ();
+    body = sum eq1.body eq2.body; constant = eq1.constant + eq2.constant }
+
+exception FACTOR1
+
+let rec chop_factor_1 = function
+  | x :: l -> 
+      if abs x.c = 1 then x,l else let (c',l') = chop_factor_1 l in (c',x::l')
+  | [] -> raise FACTOR1
+
+exception CHOPVAR
+
+let rec chop_var v = function
+  | f :: l -> if f.v = v then f,l else let (f',l') = chop_var v l in (f',f::l')
+  | [] -> raise CHOPVAR
+
+let normalize ({id=id; kind=eq_flag; body=e; constant =x} as eq) =
+  if e = [] then begin 
+    match eq_flag with
+      | EQUA ->
+          if x =0 then [] else begin
+            add_event (CONSTANT_NOT_NUL(id,x)); raise UNSOLVABLE
+         end
+      | DISE ->
+          if x <> 0 then [] else begin
+            add_event (CONSTANT_NUL id); raise UNSOLVABLE
+          end
+      | INEQ ->
+          if x >= 0 then [] else begin
+            add_event (CONSTANT_NEG(id,x)); raise UNSOLVABLE
+          end
+  end else
+    let gcd = pgcd_l (List.map (fun f -> abs f.c) e) in
+    if eq_flag=EQUA & x mod gcd <> 0 then begin
+      add_event (NOT_EXACT_DIVIDE (eq,gcd)); raise UNSOLVABLE
+    end else if eq_flag=DISE & x mod gcd <> 0 then begin
+      add_event (FORGET_C eq.id); []
+    end else if gcd <> 1 then begin
+      let c = floor_div x gcd in
+      let d = x - c * gcd in
+      let new_eq = {id=id; kind=eq_flag; constant=c; 
+                    body=map_eq_linear (fun c -> c / gcd) e} in
+      add_event (if eq_flag=EQUA or eq_flag = DISE then EXACT_DIVIDE(eq,gcd)
+                 else DIVIDE_AND_APPROX(eq,new_eq,gcd,d));
+      [new_eq]
+    end else [eq]
+
+let eliminate_with_in new_eq_id {v=v;c=c_unite} eq2
+                        ({body=e1; constant=c1} as eq1) =
+  try
+    let (f,_) = chop_var v e1 in 
+    let coeff = if c_unite=1 then -f.c else if c_unite= -1 then f.c 
+                else failwith "eliminate_with_in" in
+    let res = sum_afine new_eq_id eq1 (map_eq_afine (fun c -> c * coeff) eq2) in
+    add_event (SUM (res.id,(1,eq1),(coeff,eq2))); res
+  with CHOPVAR -> eq1
+
+let omega_mod a b = a - b * floor_div (2 * a + b) (2 * b)
+let banerjee_step (new_eq_id,new_var_id,print_var) original l1 l2 = 
+  let e = original.body in
+  let sigma = new_var_id () in
+  let smallest,var =
+    try
+      List.fold_left (fun (v,p) c ->  if v > (abs c.c) then abs c.c,c.v else (v,p))
+              (abs (List.hd e).c, (List.hd e).v) (List.tl e)
+    with Failure "tl" -> display_system print_var [original] ; failwith "TL" in
+  let m = smallest + 1 in
+  let new_eq =
+    { constant = omega_mod original.constant m;
+      body = {c= -m;v=sigma} :: 
+             map_eq_linear (fun a -> omega_mod a m) original.body;
+      id = new_eq_id (); kind = EQUA } in
+  let definition =
+    { constant = - floor_div (2 * original.constant + m) (2 * m);
+      body = map_eq_linear (fun a -> - floor_div (2 * a + m) (2 * m))
+                             original.body;
+      id = new_eq_id (); kind = EQUA } in
+  add_event (STATE {st_new_eq = new_eq; st_def = definition;
+		    st_orig =original; st_coef = m; st_var = sigma});
+  let new_eq = List.hd (normalize new_eq) in
+  let eliminated_var, def = chop_var var new_eq.body in
+  let other_equations = 
+    flat_map (fun e -> normalize (eliminate_with_in new_eq_id eliminated_var new_eq e))
+             l1 in
+  let inequations = 
+    flat_map (fun e -> normalize (eliminate_with_in new_eq_id eliminated_var new_eq e))
+             l2 in
+  let original' = eliminate_with_in new_eq_id eliminated_var new_eq original in
+  let mod_original = map_eq_afine (fun c -> c / m) original' in
+  add_event (EXACT_DIVIDE (original',m));
+  List.hd (normalize mod_original),other_equations,inequations
+
+let rec eliminate_one_equation ((new_eq_id,new_var_id,print_var) as new_ids) (e,other,ineqs) = 
+  if !debug then display_system print_var (e::other);
+  try
+    let v,def = chop_factor_1 e.body in
+    (flat_map (fun e' -> normalize (eliminate_with_in new_eq_id v e e')) other,
+     flat_map (fun e' -> normalize (eliminate_with_in new_eq_id v e e')) ineqs)
+  with FACTOR1 -> 
+    eliminate_one_equation new_ids (banerjee_step new_ids e other ineqs)
+
+let rec banerjee ((_,_,print_var) as new_ids) (sys_eq,sys_ineq) =
+  let rec fst_eq_1 = function
+     (eq::l) -> 
+        if List.exists (fun x -> abs x.c = 1) eq.body then eq,l
+        else let (eq',l') = fst_eq_1 l in (eq',eq::l')
+   | [] -> raise Not_found in
+  match sys_eq with
+     [] -> if !debug then display_system print_var sys_ineq; sys_ineq
+   | (e1::rest) -> 
+       let eq,other = try fst_eq_1 sys_eq with Not_found -> (e1,rest) in
+       if eq.body = [] then 
+         if eq.constant = 0 then begin
+           add_event (FORGET_C eq.id); banerjee new_ids (other,sys_ineq)
+         end else begin
+           add_event (CONSTANT_NOT_NUL(eq.id,eq.constant)); raise UNSOLVABLE
+         end
+       else
+	 banerjee new_ids 
+	   (eliminate_one_equation new_ids (eq,other,sys_ineq))
+
+type kind = INVERTED | NORMAL
+
+let redundancy_elimination new_eq_id system =
+  let normal = function
+     ({body=f::_} as e) when f.c < 0 ->  negate_eq e, INVERTED
+   | e -> e,NORMAL in
+  let table = Hashtbl.create 7 in
+  List.iter
+    (fun e -> 
+       let ({body=ne} as nx) ,kind = normal e in
+       if ne = [] then
+         if nx.constant < 0 then begin
+           add_event (CONSTANT_NEG(nx.id,nx.constant)); raise UNSOLVABLE
+         end else add_event (FORGET_C nx.id)
+       else
+       try 
+         let (optnormal,optinvert) = Hashtbl.find table ne in
+         let final =
+           if kind = NORMAL then begin
+             match optnormal with 
+                Some v -> 
+                  let kept =
+                    if v.constant < nx.constant 
+                    then begin add_event (FORGET (v.id,nx.id));v end
+                    else begin add_event (FORGET (nx.id,v.id));nx end in
+                  (Some(kept),optinvert)
+              | None -> Some nx,optinvert
+           end else begin
+             match optinvert with 
+                Some v ->
+                  let kept =
+                    if v.constant > nx.constant 
+                    then begin add_event (FORGET_I (v.id,nx.id));v end
+                    else begin add_event (FORGET_I (nx.id,v.id));nx end in
+                  (optnormal,Some(if v.constant > nx.constant then v else nx))
+              | None -> optnormal,Some nx
+           end in
+         begin match final with
+            (Some high, Some low) -> 
+              if high.constant < low.constant then begin
+                add_event(CONTRADICTION (high,negate_eq low));
+                raise UNSOLVABLE
+              end
+          | _ -> () end;
+         Hashtbl.remove table ne;
+         Hashtbl.add table ne final
+       with Not_found ->
+         Hashtbl.add table ne 
+           (if kind = NORMAL then (Some nx,None) else (None,Some nx)))
+    system;
+  let accu_eq = ref [] in
+  let accu_ineq = ref [] in
+  Hashtbl.iter
+    (fun p0 p1 -> match (p0,p1) with 
+       | (e, (Some x, Some y)) when x.constant = y.constant -> 
+           let id=new_eq_id () in
+           add_event (MERGE_EQ(id,x,y.id));
+           push {id=id; kind=EQUA; body=x.body; constant=x.constant} accu_eq
+       | (e, (optnorm,optinvert)) ->
+           begin match optnorm with 
+               Some x -> push x accu_ineq | _ -> () end;
+           begin match optinvert with 
+               Some x -> push (negate_eq x) accu_ineq | _ -> () end)
+    table;
+  !accu_eq,!accu_ineq
+
+exception SOLVED_SYSTEM
+
+let select_variable system =
+  let table = Hashtbl.create 7 in
+  let push v c= 
+    try let r = Hashtbl.find table v in r := max !r (abs c)
+    with Not_found -> Hashtbl.add table v (ref (abs c)) in
+  List.iter (fun {body=l} -> List.iter (fun f -> push f.v f.c) l) system;
+  let vmin,cmin = ref (-1), ref 0 in
+  let var_cpt = ref 0 in
+  Hashtbl.iter
+    (fun v ({contents =  c}) ->
+       incr var_cpt;
+       if c < !cmin or !vmin = (-1) then begin vmin := v; cmin := c end)
+    table;
+  if !var_cpt < 1 then raise SOLVED_SYSTEM;
+  !vmin
+
+let classify v system =
+  List.fold_left 
+    (fun (not_occ,below,over) eq ->
+       try let f,eq' = chop_var v eq.body in
+       if f.c >= 0 then (not_occ,((f.c,eq) :: below),over)
+       else (not_occ,below,((-f.c,eq) :: over))
+       with CHOPVAR -> (eq::not_occ,below,over))
+    ([],[],[]) system
+
+let product new_eq_id dark_shadow low high =
+  List.fold_left
+    (fun accu (a,eq1) ->
+       List.fold_left
+         (fun accu (b,eq2) ->
+            let eq = 
+              sum_afine new_eq_id (map_eq_afine (fun c -> c * b) eq1)
+                (map_eq_afine (fun c -> c * a) eq2) in
+            add_event(SUM(eq.id,(b,eq1),(a,eq2)));
+            match normalize eq with
+	      | [eq] ->
+                  let final_eq =
+                    if dark_shadow then 
+                      let delta = (a - 1) * (b - 1) in
+                      add_event(WEAKEN(eq.id,delta));
+                      {id = eq.id; kind=INEQ; body = eq.body; 
+                       constant = eq.constant - delta} 
+                    else eq
+                  in final_eq :: accu
+              | (e::_) -> failwith "Product dardk"
+              | [] -> accu)
+         accu high)
+    [] low
+
+let fourier_motzkin (_,new_eq_id,print_var) dark_shadow system =
+  let v = select_variable system in
+  let (ineq_out, ineq_low,ineq_high) = classify v system in
+  let expanded = ineq_out @ product new_eq_id dark_shadow ineq_low ineq_high in
+  if !debug then display_system print_var expanded; expanded
+
+let simplify ((new_eq_id,new_var_id,print_var) as new_ids) dark_shadow system =
+  if List.exists (fun e -> e.kind = DISE) system then 
+    failwith "disequation in simplify";
+  clear_history ();
+  List.iter (fun e -> add_event (HYP e)) system;
+  let system = flat_map normalize system in
+  let eqs,ineqs = filter (fun e -> e.kind=EQUA) system in
+  let simp_eq,simp_ineq = redundancy_elimination new_eq_id ineqs in
+  let system = (eqs @ simp_eq,simp_ineq) in
+  let rec loop1a system =
+    let sys_ineq = banerjee new_ids system in 
+    loop1b sys_ineq 
+  and loop1b sys_ineq =
+    let simp_eq,simp_ineq = redundancy_elimination new_eq_id sys_ineq in
+    if simp_eq = [] then simp_ineq else loop1a (simp_eq,simp_ineq) 
+  in
+  let rec loop2 system =
+    try
+      let expanded = fourier_motzkin new_ids dark_shadow system in
+      loop2 (loop1b expanded)
+    with SOLVED_SYSTEM ->
+      if !debug then display_system print_var system; system 
+  in
+  loop2 (loop1a system)
+
+let rec depend relie_on accu = function
+  | act :: l -> 
+      begin match act with
+	| DIVIDE_AND_APPROX (e,_,_,_) ->
+            if List.mem e.id relie_on then depend relie_on (act::accu) l
+            else depend relie_on accu l
+	| EXACT_DIVIDE (e,_) ->
+            if List.mem e.id relie_on then depend relie_on (act::accu) l
+            else depend relie_on accu l
+	| WEAKEN (e,_) ->
+            if List.mem e relie_on then depend relie_on (act::accu) l
+            else depend relie_on accu l
+	| SUM (e,(_,e1),(_,e2)) -> 
+            if List.mem e relie_on then 
+	      depend (e1.id::e2.id::relie_on) (act::accu) l
+            else 
+	      depend relie_on accu l
+	| STATE {st_new_eq=e} ->
+            if List.mem e.id relie_on then depend relie_on (act::accu) l
+            else depend relie_on accu l
+	| HYP e ->   
+            if List.mem e.id relie_on then depend relie_on (act::accu) l
+            else depend relie_on accu l
+	| FORGET_C _ -> depend relie_on accu l
+	| FORGET _ -> depend relie_on accu l
+	| FORGET_I _ -> depend relie_on accu l
+	| MERGE_EQ (e,e1,e2) ->
+            if List.mem e relie_on then 
+	      depend (e1.id::e2::relie_on) (act::accu) l
+            else 
+	      depend relie_on accu l
+	| NOT_EXACT_DIVIDE (e,_) -> depend (e.id::relie_on) (act::accu) l
+	| CONTRADICTION (e1,e2) -> 
+	    depend (e1.id::e2.id::relie_on) (act::accu) l
+	| CONSTANT_NOT_NUL (e,_) -> depend (e::relie_on) (act::accu) l
+	| CONSTANT_NEG (e,_) -> depend (e::relie_on) (act::accu) l
+	| CONSTANT_NUL e -> depend (e::relie_on) (act::accu) l
+	| NEGATE_CONTRADICT (e1,e2,_) -> 
+            depend (e1.id::e2.id::relie_on) (act::accu) l
+	| SPLIT_INEQ _ -> failwith "depend"
+      end
+  | [] -> relie_on, accu
+
+(*
+let depend relie_on accu trace = 
+  Printf.printf "Longueur de la trace initiale : %d\n" 
+    (trace_length trace + trace_length accu);
+  let rel',trace' = depend relie_on accu trace in
+  Printf.printf "Longueur de la trace simplifiée : %d\n" (trace_length trace');
+  rel',trace'
+*)
+
+let solve (new_eq_id,new_eq_var,print_var) system =
+  try let _ = simplify new_eq_id false system in failwith "no contradiction"
+  with UNSOLVABLE -> display_action print_var (snd (depend [] [] (history ())))
+      
+let negation (eqs,ineqs) =
+  let diseq,_ = filter (fun e -> e.kind = DISE) ineqs in
+  let normal = function
+    | ({body=f::_} as e) when f.c < 0 ->  negate_eq e, INVERTED
+    | e -> e,NORMAL in
+  let table = Hashtbl.create 7 in
+  List.iter (fun e -> 
+               let {body=ne;constant=c} ,kind = normal e in
+               Hashtbl.add table (ne,c) (kind,e)) diseq;
+  List.iter (fun e ->
+               if e.kind <> EQUA then pp 9999;
+               let {body=ne;constant=c},kind = normal e in
+               try 
+		 let (kind',e') = Hashtbl.find table (ne,c) in
+		 add_event (NEGATE_CONTRADICT (e,e',kind=kind'));
+		 raise UNSOLVABLE
+               with Not_found -> ()) eqs
+
+exception FULL_SOLUTION of action list * int list
+
+let simplify_strong ((new_eq_id,new_var_id,print_var) as new_ids) system =
+  clear_history ();
+  List.iter (fun e -> add_event (HYP e)) system;
+  (* Initial simplification phase *)
+  let rec loop1a system =
+    negation system;
+    let sys_ineq = banerjee new_ids system in 
+    loop1b sys_ineq 
+  and loop1b sys_ineq =
+    let dise,ine = filter (fun e -> e.kind = DISE) sys_ineq in
+    let simp_eq,simp_ineq = redundancy_elimination new_eq_id ine in
+    if simp_eq = [] then dise @ simp_ineq
+    else loop1a (simp_eq,dise @ simp_ineq) 
+  in
+  let rec loop2 system =
+    try
+      let expanded = fourier_motzkin new_ids false system in
+      loop2 (loop1b expanded)
+    with SOLVED_SYSTEM -> if !debug then display_system print_var system; system 
+  in
+  let rec explode_diseq = function 
+    | (de::diseq,ineqs,expl_map) ->
+	let id1 = new_eq_id () 
+	and id2 = new_eq_id () in
+	let e1 = 
+	  {id = id1; kind=INEQ; body = de.body; constant = de.constant - 1} in
+	let e2 = 
+	  {id = id2; kind=INEQ; body = map_eq_linear (fun x -> -x) de.body; 
+           constant = - de.constant - 1} in
+	let new_sys =
+          List.map (fun (what,sys) -> ((de.id,id1,true)::what, e1::sys)) 
+	    ineqs @ 
+          List.map (fun (what,sys) -> ((de.id,id2,false)::what,e2::sys)) 
+	    ineqs 
+	in
+	explode_diseq (diseq,new_sys,(de.id,(de,id1,id2))::expl_map)
+    | ([],ineqs,expl_map) -> ineqs,expl_map 
+  in
+  try 
+    let system = flat_map normalize system in
+    let eqs,ineqs = filter (fun e -> e.kind=EQUA) system in
+    let dise,ine = filter (fun e -> e.kind = DISE) ineqs in
+    let simp_eq,simp_ineq = redundancy_elimination new_eq_id ine in
+    let system = (eqs @ simp_eq,simp_ineq @ dise) in
+    let system' = loop1a system in
+    let diseq,ineq = filter (fun e -> e.kind = DISE) system' in
+    let first_segment = history () in
+    let sys_exploded,explode_map = explode_diseq (diseq,[[],ineq],[]) in
+    let all_solutions =
+      List.map
+        (fun (decomp,sys) -> 
+           clear_history ();
+           try let _ = loop2 sys in raise NO_CONTRADICTION
+           with UNSOLVABLE -> 
+             let relie_on,path = depend [] [] (history ()) in
+             let dc,_ = filter (fun (_,id,_) -> List.mem id relie_on) decomp in
+             let red = List.map (fun (x,_,_) -> x) dc in
+             (red,relie_on,decomp,path))
+        sys_exploded 
+    in
+    let max_count sys = 
+      let tbl = Hashtbl.create 7 in
+      let augment x = 
+        try incr (Hashtbl.find tbl x) 
+	with Not_found -> Hashtbl.add tbl x (ref 1) in
+      let eq = ref (-1) and c = ref 0 in
+      List.iter (function 
+		   | ([],r_on,_,path) -> raise (FULL_SOLUTION (path,r_on))
+                   | (l,_,_,_) -> List.iter augment l) sys;
+      Hashtbl.iter (fun x v -> if !v > !c then begin eq := x; c := !v end) tbl;
+      !eq 
+    in
+    let rec solve systems = 
+      try 
+	let id = max_count systems in 
+        let rec sign = function 
+	  | ((id',_,b)::l) -> if id=id' then b else sign l 
+          | [] -> failwith "solve" in
+        let s1,s2 = filter (fun (_,_,decomp,_) -> sign decomp) systems in
+        let s1' = 
+	  List.map (fun (dep,ro,dc,pa) -> (Util.list_except id dep,ro,dc,pa)) s1 in
+        let s2' = 
+	  List.map (fun (dep,ro,dc,pa) -> (Util.list_except id dep,ro,dc,pa)) s2 in
+        let (r1,relie1) = solve s1' 
+	and (r2,relie2) = solve s2' in
+	let (eq,id1,id2) = List.assoc id explode_map in
+	[SPLIT_INEQ(eq,(id1,r1),(id2, r2))], eq.id :: Util.list_union relie1 relie2
+      with FULL_SOLUTION (x0,x1) -> (x0,x1) 
+    in
+    let act,relie_on = solve all_solutions in
+    snd(depend relie_on act first_segment)
+  with UNSOLVABLE -> snd (depend [] [] (history ()))
diff --git a/contrib/romega/refl_omega.ml b/contrib/romega/refl_omega.ml
new file mode 100644
index 00000000..ef68c587
--- /dev/null
+++ b/contrib/romega/refl_omega.ml
@@ -0,0 +1,1307 @@
+(*************************************************************************
+
+   PROJET RNRT Calife - 2001
+   Author: Pierre Crégut - France Télécom R&D
+   Licence : LGPL version 2.1
+
+ *************************************************************************)
+
+open Const_omega
+
+
+(* \section{Useful functions and flags} *)
+(* Especially useful debugging functions *)
+let debug = ref false
+
+let show_goal gl =
+  if !debug then Pp.ppnl (Tacmach.pr_gls gl); Tacticals.tclIDTAC gl
+
+let pp i = print_int i; print_newline (); flush stdout
+
+(* More readable than the prefix notation *)
+let (>>) = Tacticals.tclTHEN
+
+(* [list_index t l = i] \eqv $nth l i = t \wedge \forall j < i nth l j != t$ *)
+
+let list_index t = 
+  let rec loop i = function
+    | (u::l) -> if u = t then i else loop (i+1) l
+    | [] -> raise Not_found in
+  loop 0
+
+(* [list_uniq l = filter_i (x i -> nth l (i-1) != x) l] *)
+let list_uniq l  = 
+  let rec uniq = function
+      x :: ((y :: _) as l) when x = y -> uniq l
+    | x :: l -> x :: uniq l
+    | [] -> [] in
+  uniq (List.sort compare l)
+
+(* $\forall x. mem x  (list\_union l1 l2) \eqv x \in \{l1\} \cup \{l2\}$ *)
+let list_union l1 l2 =
+  let rec loop buf = function
+      x :: r -> if List.mem x l2 then loop buf r else loop (x :: buf) r
+    | [] -> buf in
+  loop l2 l1
+
+(* $\forall x. 
+      mem \;\; x \;\; (list\_intersect\;\; l1\;\;l2) \eqv x \in \{l1\} 
+       \cap \{l2\}$ *)
+let list_intersect l1 l2 =
+  let rec loop buf = function
+      x :: r -> if List.mem x l2 then loop (x::buf) r else loop buf r
+    | [] -> buf in
+  loop [] l1
+
+(* cartesian product. Elements are lists and are concatenated.
+   $cartesian [x_1 ... x_n] [y_1 ... y_p] = [x_1 @ y_1, x_2 @ y_1 ... x_n @ y_1 , x_1 @ y_2 ... x_n @ y_p]$ *)
+
+let rec cartesien l1 l2 =
+  let rec loop = function
+      (x2 :: r2) -> List.map (fun x1 -> x1 @ x2) l1 @ loop r2
+    | [] -> [] in
+  loop l2
+
+(* remove element e from list l *)
+let list_remove e l =
+  let rec loop = function
+      x :: l -> if x = e then loop l else x :: loop l
+    | [] -> [] in
+  loop l
+
+(* equivalent of the map function but no element is added when the function
+   raises an exception (and the computation silently continues) *)
+let map_exc f = 
+  let rec loop = function 
+    (x::l) -> 
+      begin match try Some (f x) with exc -> None with
+        Some v -> v :: loop l | None -> loop l
+      end
+    | [] -> [] in
+  loop 
+
+let mkApp = Term.mkApp
+
+(* \section{Types}
+  \subsection{How to walk in a term}
+  To represent how to get to a proposition. Only choice points are
+  kept (branch to choose in a disjunction  and identifier of the disjunctive 
+  connector) *)
+type direction = Left of int | Right of int
+
+(* Step to find a proposition (operators are at most binary). A list is
+   a path *)
+type occ_step = O_left | O_right | O_mono
+type occ_path = occ_step list
+
+(* chemin identifiant une proposition sous forme du nom de l'hypothèse et
+   d'une liste de pas à partir de la racine de l'hypothèse *)
+type occurence = {o_hyp : Names.identifier; o_path : occ_path}
+
+(* \subsection{refiable formulas} *)
+type oformula =
+    (* integer *)
+  | Oint of int
+    (* recognized binary and unary operations *)
+  | Oplus of oformula * oformula
+  | Omult of oformula * oformula
+  | Ominus of oformula * oformula
+  | Oopp of  oformula
+    (* an atome in the environment *)
+  | Oatom of int
+    (* weird expression that cannot be translated *)
+  | Oufo of oformula
+
+(* Operators for comparison recognized by Omega *)
+type comparaison = Eq | Leq | Geq | Gt | Lt | Neq
+
+(* Type des prédicats réifiés (fragment de calcul propositionnel. Les 
+ * quantifications sont externes au langage) *)
+type oproposition = 
+    Pequa of Term.constr * oequation
+  | Ptrue 
+  | Pfalse
+  | Pnot of oproposition
+  | Por of int * oproposition * oproposition
+  | Pand of int * oproposition * oproposition
+  | Pimp of int * oproposition * oproposition
+  | Pprop of Term.constr
+
+(* Les équations ou proposiitions atomiques utiles du calcul *)
+and oequation = {
+    e_comp: comparaison;		(* comparaison *)
+    e_left: oformula;			(* formule brute gauche *)
+    e_right: oformula;			(* formule brute droite *)
+    e_trace: Term.constr;		(* tactique de normalisation *)
+    e_origin: occurence;		(* l'hypothèse dont vient le terme *)
+    e_negated: bool;			(* vrai si apparait en position nié 
+					   après normalisation *)
+    e_depends: direction  list;		(* liste des points de disjonction dont 
+					   dépend l'accès à l'équation avec la 
+					   direction (branche) pour y accéder *)
+    e_omega: Omega2.afine		(* la fonction normalisée *)
+  } 
+
+(* \subsection{Proof context} 
+  This environment codes 
+  \begin{itemize}
+  \item the terms and propositions that are given as
+  parameters of the reified proof (and are represented as variables in the
+  reified goals)
+  \item translation functions linking the decision procedure and the Coq proof
+  \end{itemize} *)
+
+type environment = {
+  (* La liste des termes non reifies constituant l'environnement global *)
+  mutable terms : Term.constr list;
+  (* La meme chose pour les propositions *)
+  mutable props : Term.constr list;
+  (* Les variables introduites par omega *)
+  mutable om_vars : (oformula * int) list;
+  (* Traduction des indices utilisés ici en les indices finaux utilisés par 
+   * la tactique Omega après dénombrement des variables utiles *)
+  real_indices : (int,int) Hashtbl.t;
+  mutable cnt_connectors : int;
+  equations : (int,oequation) Hashtbl.t;
+  constructors : (int, occurence) Hashtbl.t
+}
+
+(* \subsection{Solution tree}
+   Définition d'une solution trouvée par Omega sous la forme d'un identifiant,
+   d'un ensemble d'équation dont dépend la solution et d'une trace *)
+type solution = {
+  s_index : int;
+  s_equa_deps : int list;
+  s_trace : Omega2.action list }
+
+(* Arbre de solution résolvant complètement un ensemble de systèmes *)
+type solution_tree = 
+    Leaf of solution
+    (* un noeud interne représente un point de branchement correspondant à
+       l'élimination d'un connecteur générant plusieurs buts
+       (typ. disjonction). Le premier argument
+       est l'identifiant du connecteur *)
+  | Tree of int * solution_tree * solution_tree
+
+(* Représentation de l'environnement extrait du but initial sous forme de
+   chemins pour extraire des equations ou d'hypothèses *)
+
+type context_content = 
+    CCHyp of occurence
+  | CCEqua of int
+
+(* \section{Specific utility functions to handle base types} *)
+(* Nom arbitraire de l'hypothèse codant la négation du but final *)  
+let id_concl = Names.id_of_string "__goal__"
+
+(* Initialisation de l'environnement de réification de la tactique *)
+let new_environment () = {
+  terms = []; props = []; om_vars = []; cnt_connectors = 0; 
+  real_indices = Hashtbl.create 7;
+  equations = Hashtbl.create 7;
+  constructors = Hashtbl.create 7;
+}
+
+(* Génération d'un nom d'équation *)
+let new_eq_id env =
+   env.cnt_connectors <- env.cnt_connectors + 1; env.cnt_connectors
+
+(* Calcul de la branche complémentaire *)
+let barre = function Left x -> Right x | Right x -> Left x
+
+(* Identifiant associé à une branche *)
+let indice = function Left x | Right x -> x 
+
+(* Affichage de l'environnement de réification (termes et propositions) *)
+let print_env_reification env = 
+  let rec loop c i = function
+      [] -> Printf.printf "===============================\n\n"
+    | t :: l -> 
+	Printf.printf "(%c%02d) : " c i;
+	Pp.ppnl (Printer.prterm t);
+	Pp.flush_all ();
+	loop c (i+1) l in
+  Printf.printf "PROPOSITIONS :\n\n"; loop 'P' 0 env.props;
+  Printf.printf "TERMES :\n\n"; loop 'V' 0 env.terms
+
+
+(* \subsection{Gestion des environnements de variable pour Omega} *)
+(* generation d'identifiant d'equation pour Omega *)
+let new_omega_id = let cpt = ref 0 in function () -> incr cpt; !cpt
+(* Affichage des variables d'un système *)
+let display_omega_id i = Printf.sprintf "O%d" i
+(* Recherche la variable codant un terme pour Omega et crée la variable dans
+   l'environnement si il n'existe pas. Cas ou la variable dans Omega représente
+   le terme d'un monome (le plus souvent un atome) *)
+
+let intern_omega env t =
+  begin try List.assoc t env.om_vars
+  with Not_found -> 
+    let v = new_omega_id () in 
+    env.om_vars <- (t,v) :: env.om_vars; v
+  end
+
+(* Ajout forcé d'un lien entre un terme et une variable Omega. Cas ou la
+   variable est crée par Omega et ou il faut la lier après coup a un atome
+   réifié introduit de force *)
+let intern_omega_force env t v = env.om_vars <- (t,v) :: env.om_vars
+
+(* Récupère le terme associé à une variable *)
+let unintern_omega env id =
+  let rec loop = function 
+	[] -> failwith "unintern" 
+      | ((t,j)::l) -> if id = j then t else loop l in
+  loop env.om_vars
+
+(* \subsection{Gestion des environnements de variable pour la réflexion} 
+   Gestion des environnements de traduction entre termes des constructions
+   non réifiés et variables des termes reifies. Attention il s'agit de 
+   l'environnement initial contenant tout. Il faudra le réduire après
+   calcul des variables utiles. *)
+
+let add_reified_atom t env =
+  try list_index t env.terms
+  with Not_found ->
+    let i = List.length env.terms in
+    env.terms <- env.terms @ [t]; i
+
+let get_reified_atom env = 
+  try List.nth env.terms with _ -> failwith "get_reified_atom"
+
+(* \subsection{Gestion de l'environnement de proposition pour Omega} *)
+(* ajout d'une proposition *)
+let add_prop env t =
+  try list_index t env.props
+  with Not_found ->
+    let i = List.length env.props in  env.props <- env.props @ [t]; i
+
+(* accès a une proposition *)
+let get_prop v env = try List.nth v env with _ -> failwith "get_prop"
+
+(* \subsection{Gestion du nommage des équations} *)
+(* Ajout d'une equation dans l'environnement de reification *)
+let add_equation env e =
+  let id = e.e_omega.Omega2.id in
+  try let _ = Hashtbl.find env.equations id in ()
+  with Not_found -> Hashtbl.add env.equations id e
+
+(* accès a une equation *)
+let get_equation env id = 
+  try Hashtbl.find env.equations id
+  with e -> Printf.printf "Omega Equation %d non trouvée\n" id; raise e
+
+(* Affichage des termes réifiés *)
+let rec oprint ch = function 
+  | Oint n -> Printf.fprintf ch "%d" n
+  | Oplus (t1,t2) -> Printf.fprintf ch "(%a + %a)" oprint t1 oprint t2  
+  | Omult (t1,t2) -> Printf.fprintf ch "(%a * %a)" oprint t1 oprint t2 
+  | Ominus(t1,t2) -> Printf.fprintf ch "(%a - %a)" oprint t1 oprint t2 
+  | Oopp t1 ->Printf.fprintf ch "~ %a" oprint t1
+  | Oatom n -> Printf.fprintf ch "V%02d" n
+  | Oufo x ->  Printf.fprintf ch "?"
+
+let rec pprint ch = function
+    Pequa (_,{ e_comp=comp; e_left=t1; e_right=t2 })  ->
+      let connector = 
+	match comp with 
+	  Eq -> "=" | Leq -> "=<" | Geq -> ">="
+	| Gt -> ">" | Lt -> "<" | Neq -> "!=" in
+      Printf.fprintf ch "%a %s %a" oprint t1 connector oprint t2  
+  | Ptrue -> Printf.fprintf ch "TT"
+  | Pfalse -> Printf.fprintf ch "FF"
+  | Pnot t -> Printf.fprintf ch "not(%a)" pprint t
+  | Por (_,t1,t2) -> Printf.fprintf ch "(%a or %a)" pprint t1 pprint t2  
+  | Pand(_,t1,t2) -> Printf.fprintf ch "(%a and %a)" pprint t1 pprint t2  
+  | Pimp(_,t1,t2) -> Printf.fprintf ch "(%a => %a)" pprint t1 pprint t2  
+  | Pprop c -> Printf.fprintf ch "Prop"
+
+let rec weight env = function
+  | Oint _ -> -1
+  | Oopp c -> weight env c
+  | Omult(c,_) -> weight env c
+  | Oplus _ -> failwith "weight"
+  | Ominus _ -> failwith "weight minus"
+  | Oufo _ -> -1
+  | Oatom _ as c -> (intern_omega env c)
+
+(* \section{Passage entre oformules et représentation interne de Omega} *)
+
+(* \subsection{Oformula vers Omega} *)
+
+let omega_of_oformula env kind = 
+  let rec loop accu = function
+    | Oplus(Omult(v,Oint n),r) -> 
+	loop ({Omega2.v=intern_omega env v; Omega2.c=n} :: accu) r
+    | Oint n ->
+	let id = new_omega_id () in
+	(*i tag_equation name id; i*)
+	{Omega2.kind = kind; Omega2.body = List.rev accu; 
+	 Omega2.constant = n; Omega2.id = id}
+    | t -> print_string "CO"; oprint stdout t; failwith "compile_equation" in
+  loop []
+
+(* \subsection{Omega vers Oformula} *)
+
+let reified_of_atom env i =
+  try Hashtbl.find env.real_indices i
+  with Not_found -> 
+    Printf.printf "Atome %d non trouvé\n" i; 
+    Hashtbl.iter (fun k v -> Printf.printf "%d -> %d\n" k v) env.real_indices;
+    raise Not_found
+
+let rec oformula_of_omega env af = 
+  let rec loop = function
+    | ({Omega2.v=v; Omega2.c=n}::r) ->
+	Oplus(Omult(unintern_omega env v,Oint n),loop r)
+    | [] -> Oint af.Omega2.constant in
+  loop af.Omega2.body
+
+let app f v = mkApp(Lazy.force f,v)
+
+(* \subsection{Oformula vers COQ reel} *)
+
+let rec coq_of_formula env t =
+  let rec loop = function
+  | Oplus (t1,t2) -> app coq_Zplus [| loop t1; loop t2 |]
+  | Oopp t -> app coq_Zopp [| loop t |]
+  | Omult(t1,t2) -> app coq_Zmult [| loop t1; loop t2 |]
+  | Oint v ->   mk_Z v
+  | Oufo t -> loop t
+  | Oatom var ->
+      (* attention ne traite pas les nouvelles variables si on ne les
+       * met pas dans env.term *)
+      get_reified_atom env var
+  | Ominus(t1,t2) -> app coq_Zminus [| loop t1; loop t2 |] in
+  loop t
+
+(* \subsection{Oformula vers COQ reifié} *)
+
+let rec reified_of_formula env = function
+  | Oplus (t1,t2) ->
+      app coq_t_plus [| reified_of_formula env t1; reified_of_formula env t2 |]
+  | Oopp t ->
+      app coq_t_opp [| reified_of_formula env t |]
+  | Omult(t1,t2) ->
+      app coq_t_mult [| reified_of_formula env t1; reified_of_formula env t2 |]
+  | Oint v ->  app coq_t_int [| mk_Z v |]
+  | Oufo t -> reified_of_formula env t
+  | Oatom i -> app coq_t_var [| mk_nat (reified_of_atom env i) |]
+  | Ominus(t1,t2) ->
+      app coq_t_minus [| reified_of_formula env t1; reified_of_formula env t2 |]
+
+let reified_of_formula env f =
+  begin try reified_of_formula env f with e -> oprint stderr f; raise e end
+
+let rec reified_of_proposition env = function
+    Pequa (_,{ e_comp=Eq; e_left=t1; e_right=t2 }) -> 
+      app coq_p_eq [| reified_of_formula env t1; reified_of_formula env t2 |]
+  | Pequa (_,{ e_comp=Leq; e_left=t1; e_right=t2 }) -> 
+      app coq_p_leq [| reified_of_formula env t1; reified_of_formula env t2 |]
+  | Pequa(_,{ e_comp=Geq; e_left=t1; e_right=t2 }) -> 
+      app coq_p_geq [| reified_of_formula env t1; reified_of_formula env t2 |]
+  | Pequa(_,{ e_comp=Gt; e_left=t1; e_right=t2 }) -> 
+      app coq_p_gt [| reified_of_formula env t1; reified_of_formula env t2 |]
+  | Pequa(_,{ e_comp=Lt; e_left=t1; e_right=t2 }) ->  
+      app coq_p_lt [| reified_of_formula env t1; reified_of_formula env t2 |]
+  | Pequa(_,{ e_comp=Neq; e_left=t1; e_right=t2 }) -> 
+      app coq_p_neq [| reified_of_formula env t1; reified_of_formula env t2 |]
+  | Ptrue -> Lazy.force coq_p_true
+  | Pfalse -> Lazy.force coq_p_false
+  | Pnot t -> 
+      app coq_p_not [| reified_of_proposition env t |]
+  | Por (_,t1,t2) -> 
+      app coq_p_or
+	[| reified_of_proposition env t1; reified_of_proposition env t2 |]
+  | Pand(_,t1,t2) -> 
+      app coq_p_and
+	[| reified_of_proposition env t1; reified_of_proposition env t2 |]
+  | Pimp(_,t1,t2) -> 
+      app coq_p_imp
+	[| reified_of_proposition env t1; reified_of_proposition env t2 |]
+  | Pprop t -> app coq_p_prop [| mk_nat (add_prop env t) |]
+
+let reified_of_proposition env f =
+  begin try reified_of_proposition env f 
+  with e -> pprint stderr f; raise e end
+
+(* \subsection{Omega vers COQ réifié} *)
+
+let reified_of_omega env body constant = 
+  let coeff_constant = 
+    app coq_t_int [| mk_Z constant |] in
+  let mk_coeff {Omega2.c=c; Omega2.v=v} t =
+    let coef = 
+      app coq_t_mult 
+	[| reified_of_formula env (unintern_omega env v); 
+	   app coq_t_int [| mk_Z c |] |] in
+    app coq_t_plus [|coef; t |] in
+  List.fold_right mk_coeff body coeff_constant
+
+let reified_of_omega env body c  = 
+  begin try 
+    reified_of_omega env body c 
+  with e -> 
+    Omega2.display_eq display_omega_id (body,c); raise e 
+  end
+
+(* \section{Opérations sur les équations}
+Ces fonctions préparent les traces utilisées par la tactique réfléchie
+pour faire des opérations de normalisation sur les équations.  *)
+
+(* \subsection{Extractions des variables d'une équation} *)
+(* Extraction des variables d'une équation *)
+
+let rec vars_of_formula = function
+  | Oint _ -> []
+  | Oplus (e1,e2)  -> (vars_of_formula e1) @ (vars_of_formula e2)
+  | Omult (e1,e2)  -> (vars_of_formula e1) @ (vars_of_formula e2)
+  | Ominus (e1,e2) -> (vars_of_formula e1) @ (vars_of_formula e2)
+  | Oopp e -> (vars_of_formula e)
+  | Oatom i -> [i]
+  | Oufo _ -> []
+
+let vars_of_equations l =
+  let rec loop = function
+      e :: l -> vars_of_formula e.e_left @ vars_of_formula e.e_right @ loop l
+    | [] -> [] in
+  list_uniq (List.sort compare (loop l))
+
+(* \subsection{Multiplication par un scalaire} *)
+
+let rec scalar n = function
+   Oplus(t1,t2) -> 
+     let tac1,t1' = scalar  n t1 and 
+         tac2,t2' = scalar  n t2 in
+     do_list [Lazy.force coq_c_mult_plus_distr; do_both tac1 tac2], 
+     Oplus(t1',t2')
+ | Oopp t ->
+     do_list [Lazy.force coq_c_mult_opp_left], Omult(t,Oint(-n))
+ | Omult(t1,Oint x) -> 
+     do_list [Lazy.force coq_c_mult_assoc_reduced], Omult(t1,Oint (n*x))
+ | Omult(t1,t2) -> 
+     Util.error "Omega: Can't solve a goal with non-linear products"
+ | (Oatom _ as t) -> do_list [], Omult(t,Oint n)
+ | Oint i -> do_list [Lazy.force coq_c_reduce],Oint(n*i)
+ | (Oufo _ as t)-> do_list [], Oufo (Omult(t,Oint n))
+ | Ominus _ -> failwith "scalar minus"
+
+(* \subsection{Propagation de l'inversion} *)
+
+let rec negate = function
+   Oplus(t1,t2) -> 
+     let tac1,t1' = negate t1 and 
+         tac2,t2' = negate t2 in
+     do_list [Lazy.force coq_c_opp_plus ; (do_both tac1 tac2)],
+     Oplus(t1',t2')
+ | Oopp t ->
+     do_list [Lazy.force coq_c_opp_opp], t
+ | Omult(t1,Oint x) -> 
+     do_list [Lazy.force coq_c_opp_mult_r], Omult(t1,Oint (-x))
+ | Omult(t1,t2) -> 
+     Util.error "Omega: Can't solve a goal with non-linear products"
+ | (Oatom _ as t) ->
+     do_list [Lazy.force coq_c_opp_one], Omult(t,Oint(-1))
+ | Oint i -> do_list [Lazy.force coq_c_reduce] ,Oint(-i)
+ | Oufo c -> do_list [], Oufo (Oopp c)
+ | Ominus _ -> failwith "negate minus"
+
+let rec norm l = (List.length l)
+
+(* \subsection{Mélange (fusion) de deux équations} *)
+(* \subsubsection{Version avec coefficients} *)
+let rec shuffle_path k1 e1 k2 e2 =
+  let rec loop = function
+      (({Omega2.c=c1;Omega2.v=v1}::l1) as l1'),
+      (({Omega2.c=c2;Omega2.v=v2}::l2) as l2') ->
+	if v1 = v2 then
+          if k1*c1 + k2 * c2 = 0 then (
+            Lazy.force coq_f_cancel :: loop (l1,l2))
+          else (
+            Lazy.force coq_f_equal :: loop (l1,l2) )
+	else if v1 > v2 then (
+	  Lazy.force coq_f_left :: loop(l1,l2'))
+	else (
+	  Lazy.force coq_f_right :: loop(l1',l2))
+    | ({Omega2.c=c1;Omega2.v=v1}::l1), [] -> 
+	  Lazy.force coq_f_left :: loop(l1,[])
+    | [],({Omega2.c=c2;Omega2.v=v2}::l2) -> 
+	  Lazy.force coq_f_right :: loop([],l2)
+    | [],[] -> flush stdout; [] in
+  mk_shuffle_list (loop (e1,e2))
+
+(* \subsubsection{Version sans coefficients} *)
+let rec shuffle env (t1,t2) = 
+  match t1,t2 with
+    Oplus(l1,r1), Oplus(l2,r2) ->
+      if weight env l1 > weight env l2 then 
+	let l_action,t' = shuffle env (r1,t2) in
+	do_list [Lazy.force coq_c_plus_assoc_r;do_right l_action], Oplus(l1,t')
+      else 
+	let l_action,t' = shuffle env (t1,r2) in
+	do_list [Lazy.force coq_c_plus_permute;do_right l_action], Oplus(l2,t')
+  | Oplus(l1,r1), t2 -> 
+      if weight env l1 > weight env t2 then
+        let (l_action,t') = shuffle env (r1,t2) in
+	do_list [Lazy.force coq_c_plus_assoc_r;do_right l_action],Oplus(l1, t')
+      else do_list [Lazy.force coq_c_plus_sym], Oplus(t2,t1)
+  | t1,Oplus(l2,r2) -> 
+      if weight env l2 > weight env t1 then
+        let (l_action,t') = shuffle env (t1,r2) in
+	do_list [Lazy.force coq_c_plus_permute;do_right l_action], Oplus(l2,t')
+      else do_list [],Oplus(t1,t2)
+  | Oint t1,Oint t2 ->
+      do_list [Lazy.force coq_c_reduce], Oint(t1+t2)
+  | t1,t2 ->
+      if weight env t1 < weight env t2 then
+	do_list [Lazy.force coq_c_plus_sym], Oplus(t2,t1)
+      else do_list [],Oplus(t1,t2)
+
+(* \subsection{Fusion avec réduction} *)
+
+let shrink_pair f1 f2 =
+  begin match f1,f2 with
+     Oatom v,Oatom _ -> 
+       Lazy.force coq_c_red1, Omult(Oatom v,Oint 2)
+   | Oatom v, Omult(_,c2) -> 
+       Lazy.force coq_c_red2, Omult(Oatom v,Oplus(c2,Oint 1))
+   | Omult (v1,c1),Oatom v -> 
+       Lazy.force coq_c_red3, Omult(Oatom v,Oplus(c1,Oint 1))
+   | Omult (Oatom v,c1),Omult (v2,c2) ->
+       Lazy.force coq_c_red4, Omult(Oatom v,Oplus(c1,c2))
+   | t1,t2 -> 
+       oprint stdout t1; print_newline (); oprint stdout t2; print_newline (); 
+       flush Pervasives.stdout; Util.error "shrink.1"
+  end
+
+(* \subsection{Calcul d'une sous formule constante} *)
+
+let reduce_factor = function
+   Oatom v ->
+     let r = Omult(Oatom v,Oint 1) in
+       [Lazy.force coq_c_red0],r
+  | Omult(Oatom v,Oint n) as f -> [],f
+  | Omult(Oatom v,c) ->
+      let rec compute = function
+          Oint n -> n
+	| Oplus(t1,t2) -> compute t1 + compute t2 
+	| _ -> Util.error "condense.1" in
+	[Lazy.force coq_c_reduce], Omult(Oatom v,Oint(compute c))
+  | t -> Util.error "reduce_factor.1"
+
+(* \subsection{Réordonancement} *)
+
+let rec condense env = function
+    Oplus(f1,(Oplus(f2,r) as t)) ->
+      if weight env f1 = weight env f2 then begin
+	let shrink_tac,t = shrink_pair f1 f2 in
+	let assoc_tac = Lazy.force coq_c_plus_assoc_l in
+	let tac_list,t' = condense env (Oplus(t,r)) in
+	assoc_tac :: do_left (do_list [shrink_tac]) :: tac_list, t'
+      end else begin
+	let tac,f = reduce_factor f1 in
+	let tac',t' = condense env t in 
+	[do_both (do_list tac) (do_list tac')], Oplus(f,t') 
+      end
+  | (Oplus(f1,Oint n) as t) -> 
+      let tac,f1' = reduce_factor f1 in 
+      [do_left (do_list tac)],Oplus(f1',Oint n)
+  | Oplus(f1,f2) -> 
+      if weight env f1 = weight env f2 then begin
+	let tac_shrink,t = shrink_pair f1 f2 in
+	let tac,t' = condense env t in
+	tac_shrink :: tac,t'
+      end else begin
+	let tac,f = reduce_factor f1 in
+	let tac',t' = condense env f2 in 
+	[do_both (do_list tac) (do_list tac')],Oplus(f,t') 
+      end
+  | (Oint _ as t)-> [],t
+  | t -> 
+      let tac,t' = reduce_factor t in
+      let final = Oplus(t',Oint 0) in
+      tac @ [Lazy.force coq_c_red6], final
+
+(* \subsection{Elimination des zéros} *)
+
+let rec clear_zero = function
+   Oplus(Omult(Oatom v,Oint 0),r) ->
+     let tac',t = clear_zero r in
+     Lazy.force coq_c_red5 :: tac',t
+  | Oplus(f,r) -> 
+      let tac,t = clear_zero r in 
+      (if tac = [] then [] else [do_right (do_list tac)]),Oplus(f,t)
+  | t -> [],t;;
+
+(* \subsection{Transformation des hypothèses} *)
+
+let rec reduce env = function
+    Oplus(t1,t2) ->
+      let t1', trace1 = reduce env t1 in
+      let t2', trace2 = reduce env t2 in
+      let trace3,t' = shuffle env (t1',t2') in
+      t', do_list [do_both trace1 trace2; trace3]
+  | Ominus(t1,t2) ->
+      let t,trace = reduce env (Oplus(t1, Oopp t2)) in
+      t, do_list [Lazy.force coq_c_minus; trace]
+  | Omult(t1,t2) as t ->
+      let t1', trace1 = reduce env t1 in
+      let t2', trace2 = reduce env t2 in
+      begin match t1',t2' with
+      | (_, Oint n) ->
+	  let tac,t' = scalar n t1' in
+	  t', do_list [do_both trace1 trace2; tac]
+      | (Oint n,_) ->
+	  let tac,t' = scalar n t2' in
+	  t', do_list [do_both trace1 trace2; Lazy.force coq_c_mult_sym; tac]
+      | _ -> Oufo t, Lazy.force coq_c_nop
+      end
+  | Oopp t ->
+      let t',trace = reduce env  t in
+      let trace',t'' = negate t' in
+      t'', do_list [do_left trace; trace']
+  | (Oint _ | Oatom _ | Oufo _) as t -> t, Lazy.force coq_c_nop
+
+let normalize_linear_term env t =
+  let t1,trace1 = reduce env t in
+  let trace2,t2 = condense env t1 in
+  let trace3,t3 = clear_zero t2 in
+  do_list [trace1; do_list trace2; do_list trace3], t3
+
+(* Cette fonction reproduit très exactement le comportement de [p_invert] *)
+let negate_oper = function
+    Eq -> Neq | Neq -> Eq | Leq -> Gt | Geq -> Lt | Lt -> Geq | Gt -> Leq
+ 
+let normalize_equation env (negated,depends,origin,path) (oper,t1,t2) = 
+  let mk_step t1 t2 f kind =
+    let t = f t1 t2 in
+    let trace, oterm = normalize_linear_term env t in
+    let equa = omega_of_oformula env kind oterm in 
+    { e_comp = oper; e_left = t1; e_right = t2; 
+      e_negated = negated; e_depends = depends; 
+      e_origin = { o_hyp = origin; o_path = List.rev path };
+      e_trace = trace; e_omega = equa } in
+  try match (if negated then (negate_oper oper) else oper) with
+    | Eq  -> mk_step t1 t2 (fun o1 o2 -> Oplus (o1,Oopp o2)) Omega2.EQUA
+    | Neq -> mk_step t1 t2 (fun o1 o2 -> Oplus (o1,Oopp o2)) Omega2.DISE
+    | Leq -> mk_step t1 t2 (fun o1 o2 -> Oplus (o2,Oopp o1)) Omega2.INEQ
+    | Geq -> mk_step t1 t2 (fun o1 o2 -> Oplus (o1,Oopp o2)) Omega2.INEQ
+    | Lt  ->
+        mk_step t1 t2 (fun o1 o2 -> Oplus (Oplus(o2,Oint (-1)),Oopp o1))
+	  Omega2.INEQ
+    | Gt ->
+        mk_step t1 t2 (fun o1 o2 -> Oplus (Oplus(o1,Oint (-1)),Oopp o2)) 
+	  Omega2.INEQ
+  with e when Logic.catchable_exception e -> raise e
+
+(* \section{Compilation des hypothèses} *)
+
+let rec oformula_of_constr env t =
+  try match destructurate t with
+  | Kapp("Zplus",[t1;t2]) -> binop env (fun x y -> Oplus(x,y)) t1 t2
+  | Kapp("Zminus",[t1;t2]) ->binop env (fun x y -> Ominus(x,y)) t1 t2
+  | Kapp("Zmult",[t1;t2]) ->binop env (fun x y -> Omult(x,y)) t1 t2
+  | Kapp(("Zpos"|"Zneg"|"Z0"),_) -> 
+      begin try Oint(recognize_number t)
+      with _ -> Oatom (add_reified_atom t env) end
+  | _ -> 
+      Oatom (add_reified_atom t env)
+  with e when Logic.catchable_exception e ->
+    Oatom (add_reified_atom t env)
+
+and  binop env c t1 t2 = 
+    let t1' = oformula_of_constr env t1 in
+    let t2' = oformula_of_constr env t2 in
+    c t1' t2'
+
+and  binprop env (neg2,depends,origin,path) 
+             add_to_depends neg1 gl c t1 t2 =
+  let i = new_eq_id env in
+  let depends1 = if add_to_depends then Left i::depends else depends in
+  let depends2 = if add_to_depends then Right i::depends else depends in
+  if add_to_depends then
+    Hashtbl.add env.constructors i {o_hyp = origin; o_path = List.rev path};
+  let t1' = 
+    oproposition_of_constr env (neg1,depends1,origin,O_left::path) gl t1 in
+  let t2' =
+    oproposition_of_constr env (neg2,depends2,origin,O_right::path) gl t2 in
+  (* On numérote le connecteur dans l'environnement. *)
+  c i t1' t2'
+
+and mk_equation env ctxt c connector t1 t2 =
+  let t1' = oformula_of_constr env t1 in
+  let t2' = oformula_of_constr env t2 in
+  (* On ajoute l'equation dans l'environnement. *)
+  let omega = normalize_equation env ctxt (connector,t1',t2') in
+  add_equation env omega;
+  Pequa (c,omega)
+
+and oproposition_of_constr env ((negated,depends,origin,path) as ctxt) gl c = 
+  try match destructurate c with
+    | Kapp("eq",[typ;t1;t2]) 
+      when destructurate (Tacmach.pf_nf gl typ) = Kapp("Z",[]) ->
+         mk_equation env ctxt c Eq t1 t2
+    | Kapp("Zne",[t1;t2]) -> 
+	mk_equation env ctxt c Neq t1 t2
+    | Kapp("Zle",[t1;t2]) -> 
+	mk_equation env ctxt c Leq t1 t2
+    | Kapp("Zlt",[t1;t2]) -> 
+	mk_equation env ctxt c Lt t1 t2
+    | Kapp("Zge",[t1;t2]) -> 
+	mk_equation env ctxt c Geq t1 t2
+    | Kapp("Zgt",[t1;t2]) -> 
+	mk_equation env ctxt c Gt t1 t2
+    | Kapp("True",[]) -> Ptrue
+    | Kapp("False",[]) -> Pfalse
+    | Kapp("not",[t]) ->
+       let t' = 
+	 oproposition_of_constr 
+	   env (not negated, depends, origin,(O_mono::path)) gl t in 
+       Pnot t'
+    | Kapp("or",[t1;t2]) ->  
+	binprop env ctxt (not negated) negated gl (fun i x y -> Por(i,x,y)) t1 t2
+    | Kapp("and",[t1;t2]) ->
+	binprop env ctxt negated negated gl
+	  (fun i x y -> Pand(i,x,y)) t1 t2
+    | Kimp(t1,t2) ->
+	binprop env ctxt (not negated) (not negated) gl 
+	  (fun i x y -> Pimp(i,x,y)) t1 t2
+    | _ -> Pprop c
+  with e when Logic.catchable_exception e -> Pprop c
+
+(* Destructuration des hypothèses et de la conclusion *)
+
+let reify_gl env gl =
+  let concl = Tacmach.pf_concl gl in
+  let t_concl =
+    Pnot (oproposition_of_constr env (true,[],id_concl,[O_mono]) gl concl) in
+  if !debug then begin
+    Printf.printf "CONCL: "; pprint stdout t_concl; Printf.printf "\n"
+  end;
+  let rec loop = function
+      (i,t) :: lhyps ->
+	let t' = oproposition_of_constr env (false,[],i,[]) gl t in
+	if !debug then begin
+	  Printf.printf "%s: " (Names.string_of_id i);
+	  pprint stdout t';
+	  Printf.printf "\n"
+	end;
+	(i,t') :: loop lhyps
+    | [] -> 
+	if !debug then print_env_reification env; 
+	[] in
+  let t_lhyps = loop (Tacmach.pf_hyps_types gl) in
+  (id_concl,t_concl) :: t_lhyps 
+
+let rec destructurate_pos_hyp orig list_equations list_depends = function
+  | Pequa (_,e) -> [e :: list_equations]
+  | Ptrue | Pfalse | Pprop _ -> [list_equations]
+  | Pnot t -> destructurate_neg_hyp orig list_equations list_depends t
+  | Por (i,t1,t2) -> 
+      let s1 = 
+	destructurate_pos_hyp orig list_equations (i::list_depends) t1 in
+      let s2 = 
+	destructurate_pos_hyp orig list_equations (i::list_depends) t2 in
+      s1 @ s2
+  | Pand(i,t1,t2) -> 
+      let list_s1 =
+	destructurate_pos_hyp orig list_equations (list_depends) t1 in
+      let rec loop = function 
+	  le1 :: ll -> destructurate_pos_hyp orig le1 list_depends t2 @ loop ll
+	| [] -> [] in
+      loop list_s1
+  | Pimp(i,t1,t2) -> 
+      let s1 =
+	destructurate_neg_hyp orig list_equations (i::list_depends) t1 in
+      let s2 =
+	destructurate_pos_hyp orig list_equations (i::list_depends) t2 in
+      s1 @ s2
+
+and destructurate_neg_hyp orig list_equations list_depends = function
+  | Pequa (_,e) -> [e :: list_equations]
+  | Ptrue | Pfalse | Pprop _ -> [list_equations]
+  | Pnot t -> destructurate_pos_hyp orig list_equations list_depends t
+  | Pand (i,t1,t2) -> 
+      let s1 =
+	destructurate_neg_hyp orig list_equations (i::list_depends) t1 in
+      let s2 =
+	destructurate_neg_hyp orig list_equations (i::list_depends) t2 in
+      s1 @ s2
+  | Por(_,t1,t2) -> 
+      let list_s1 =
+	destructurate_neg_hyp orig list_equations list_depends t1 in
+      let rec loop = function 
+	  le1 :: ll -> destructurate_neg_hyp orig le1 list_depends t2 @ loop ll
+	| [] -> [] in
+      loop list_s1
+  | Pimp(_,t1,t2) -> 
+      let list_s1 =
+	destructurate_pos_hyp orig list_equations list_depends t1 in
+      let rec loop = function 
+	  le1 :: ll -> destructurate_neg_hyp orig le1 list_depends t2 @ loop ll
+	| [] -> [] in
+      loop list_s1
+
+let destructurate_hyps syst =
+  let rec loop = function
+      (i,t) :: l -> 
+	let l_syst1 = destructurate_pos_hyp i []  [] t in
+	let l_syst2 = loop l in
+	cartesien l_syst1 l_syst2
+    | [] -> [[]] in
+  loop syst
+
+(* \subsection{Affichage d'un système d'équation} *)
+
+(* Affichage des dépendances de système *)
+let display_depend = function
+    Left i -> Printf.printf " L%d" i 
+  | Right i -> Printf.printf " R%d" i
+
+let display_systems syst_list = 
+  let display_omega om_e = 
+    Printf.printf "%d : %a %s 0\n"
+      om_e.Omega2.id
+      (fun _ -> Omega2.display_eq display_omega_id) 
+      (om_e.Omega2.body, om_e.Omega2.constant)
+      (Omega2.operator_of_eq om_e.Omega2.kind) in
+
+  let display_equation oformula_eq = 
+    pprint stdout (Pequa (Lazy.force coq_c_nop,oformula_eq)); print_newline ();
+    display_omega oformula_eq.e_omega;
+    Printf.printf "  Depends on:"; 
+    List.iter display_depend oformula_eq.e_depends;
+    Printf.printf "\n  Path: %s" 
+      (String.concat ""
+	 (List.map (function O_left -> "L" | O_right -> "R" | O_mono -> "M")
+	    oformula_eq.e_origin.o_path));
+    Printf.printf "\n  Origin: %s -- Negated : %s\n"
+      (Names.string_of_id oformula_eq.e_origin.o_hyp)
+      (if oformula_eq.e_negated then "yes" else "false") in
+
+  let display_system syst =
+    Printf.printf "=SYSTEME==================================\n";
+    List.iter display_equation syst in
+  List.iter display_system syst_list
+
+(* Extraction des prédicats utilisées dans une trace. Permet ensuite le
+   calcul des hypothèses *)
+
+let rec hyps_used_in_trace = function
+  | act :: l -> 
+      begin match act with
+	| Omega2.HYP e -> e.Omega2.id :: hyps_used_in_trace l
+	| Omega2.SPLIT_INEQ (_,(_,act1),(_,act2)) -> 
+	    hyps_used_in_trace act1 @ hyps_used_in_trace act2
+	| _ -> hyps_used_in_trace l
+      end
+  | [] -> []
+
+(* Extraction des variables déclarées dans une équation. Permet ensuite
+   de les déclarer dans l'environnement de la procédure réflexive et
+   éviter les créations de variable au vol *)
+
+let rec variable_stated_in_trace = function
+  | act :: l -> 
+      begin match act with
+	| Omega2.STATE action ->
+	    (*i nlle_equa: afine, def: afine, eq_orig: afine, i*)
+            (*i coef: int, var:int                            i*)
+	    action :: variable_stated_in_trace l
+	| Omega2.SPLIT_INEQ (_,(_,act1),(_,act2)) -> 
+	    variable_stated_in_trace act1 @ variable_stated_in_trace act2
+	| _ -> variable_stated_in_trace l
+      end
+  | [] -> []
+;;
+
+let add_stated_equations env tree = 
+  let rec loop = function
+      Tree(_,t1,t2) -> 
+	list_union (loop t1) (loop t2)
+    | Leaf s -> variable_stated_in_trace s.s_trace in
+  (* Il faut trier les variables par ordre d'introduction pour ne pas risquer
+     de définir dans le mauvais ordre *)
+  let stated_equations = 
+    List.sort (fun x y -> x.Omega2.st_var - y.Omega2.st_var) (loop tree) in
+  let add_env st = 
+    (* On retransforme la définition de v en formule reifiée *)
+    let v_def = oformula_of_omega env st.Omega2.st_def in
+    (* Notez que si l'ordre de création des variables n'est pas respecté, 
+     * ca va planter *)
+    let coq_v = coq_of_formula env v_def in
+    let v = add_reified_atom coq_v env in
+    (* Le terme qu'il va falloir introduire *)
+    let term_to_generalize = app coq_refl_equal [|Lazy.force coq_Z; coq_v|] in
+    (* sa représentation sous forme d'équation mais non réifié car on n'a pas
+     * l'environnement pour le faire correctement *)
+    let term_to_reify = (v_def,Oatom v) in
+    (* enregistre le lien entre la variable omega et la variable Coq *)
+    intern_omega_force env (Oatom v) st.Omega2.st_var; 
+    (v, term_to_generalize,term_to_reify,st.Omega2.st_def.Omega2.id) in
+ List.map add_env stated_equations
+
+(* Calcule la liste des éclatements à réaliser sur les hypothèses 
+   nécessaires pour extraire une liste d'équations donnée *)
+
+let rec get_eclatement env = function
+    i :: r -> 
+      let l = try (get_equation env i).e_depends with Not_found -> [] in
+      list_union l (get_eclatement env r)
+  | [] -> []
+
+let select_smaller l = 
+  let comp (_,x) (_,y) = List.length x - List.length y in
+  try List.hd (List.sort comp l) with Failure _ -> failwith "select_smaller"
+
+let filter_compatible_systems required systems =
+  let rec select = function
+      (x::l) -> 
+	if List.mem x required then select l
+	else if List.mem (barre x) required then raise Exit 
+	else x :: select l
+    | [] -> [] in
+  map_exc (function (sol,splits) -> (sol,select splits)) systems
+
+let rec equas_of_solution_tree = function
+    Tree(_,t1,t2) -> 
+      list_union (equas_of_solution_tree t1) (equas_of_solution_tree t2)
+  | Leaf s -> s.s_equa_deps
+
+
+let really_useful_prop l_equa c =
+  let rec real_of = function
+      Pequa(t,_) -> t
+    | Ptrue ->  app  coq_true [||]
+    | Pfalse -> app  coq_false [||]
+    | Pnot t1 -> app coq_not [|real_of t1|]
+    | Por(_,t1,t2) -> app coq_or [|real_of t1; real_of t2|]
+    | Pand(_,t1,t2) -> app coq_and [|real_of t1; real_of t2|]
+    (* Attention : implications sur le lifting des variables à comprendre ! *)
+    | Pimp(_,t1,t2) -> Term.mkArrow (real_of t1) (real_of t2)
+    | Pprop t -> t in
+  let rec loop c = 
+    match c with 
+      Pequa(_,e) ->
+	if List.mem e.e_omega.Omega2.id l_equa then Some c else None
+    | Ptrue -> None
+    | Pfalse -> None
+    | Pnot t1 -> 
+	begin match loop t1 with None -> None | Some t1' -> Some (Pnot t1') end
+    | Por(i,t1,t2) -> binop (fun (t1,t2) -> Por(i,t1,t2)) t1 t2
+    | Pand(i,t1,t2) -> binop (fun (t1,t2) -> Pand(i,t1,t2)) t1 t2
+    | Pimp(i,t1,t2) -> binop (fun (t1,t2) -> Pimp(i,t1,t2)) t1 t2
+    | Pprop t -> None
+  and binop f t1 t2 = 
+    begin match loop t1, loop t2 with
+      None, None -> None 
+    | Some t1',Some t2' -> Some (f(t1',t2'))
+    | Some t1',None -> Some (f(t1',Pprop (real_of t2)))
+    | None,Some t2' -> Some (f(Pprop (real_of t1),t2'))
+    end in
+  match loop c with
+    None -> Pprop (real_of c)
+  | Some t -> t
+
+let rec display_solution_tree ch = function
+    Leaf t -> 
+      output_string ch 
+       (Printf.sprintf "%d[%s]" 
+	 t.s_index
+         (String.concat " " (List.map string_of_int t.s_equa_deps)))
+  | Tree(i,t1,t2) -> 
+      Printf.fprintf ch "S%d(%a,%a)" i 
+	display_solution_tree t1 display_solution_tree t2
+
+let rec solve_with_constraints all_solutions path = 
+  let rec build_tree sol buf = function
+      [] -> Leaf sol
+    | (Left i :: remainder) -> 
+	Tree(i,
+	     build_tree sol (Left i :: buf) remainder, 
+	     solve_with_constraints all_solutions (List.rev(Right i :: buf)))
+    | (Right i :: remainder) -> 
+	 Tree(i,
+	      solve_with_constraints all_solutions (List.rev (Left i ::  buf)),
+	      build_tree sol (Right i :: buf) remainder) in
+  let weighted = filter_compatible_systems path all_solutions in
+  let (winner_sol,winner_deps) =
+    try select_smaller weighted 
+    with e ->  
+      Printf.printf "%d - %d\n" 
+	(List.length weighted) (List.length all_solutions);
+      List.iter display_depend path; raise e in
+  build_tree winner_sol (List.rev path) winner_deps 
+
+let find_path {o_hyp=id;o_path=p} env = 
+  let rec loop_path = function
+      ([],l) -> Some l
+    | (x1::l1,x2::l2) when x1 = x2 -> loop_path (l1,l2)
+    | _ -> None in
+  let rec loop_id i = function
+      CCHyp{o_hyp=id';o_path=p'} :: l when id = id' ->
+	begin match loop_path (p',p) with
+	  Some r -> i,r
+	| None -> loop_id (i+1) l
+	end
+    | _ :: l -> loop_id (i+1) l
+    | [] -> failwith "find_path" in
+  loop_id 0 env
+
+let mk_direction_list l = 
+  let trans = function 
+      O_left -> coq_d_left | O_right -> coq_d_right | O_mono -> coq_d_mono in
+  mk_list (Lazy.force coq_direction) (List.map (fun d-> Lazy.force(trans d)) l)
+
+
+(* \section{Rejouer l'historique} *)
+
+let get_hyp env_hyp i =
+  try list_index (CCEqua i) env_hyp
+  with Not_found -> failwith (Printf.sprintf "get_hyp %d" i)
+
+let replay_history env env_hyp =
+  let rec loop env_hyp t =
+    match t with
+      | Omega2.CONTRADICTION (e1,e2) :: l -> 
+	  let trace = mk_nat (List.length e1.Omega2.body) in
+	  mkApp (Lazy.force coq_s_contradiction,
+		 [| trace ; mk_nat (get_hyp env_hyp e1.Omega2.id); 
+		    mk_nat (get_hyp env_hyp e2.Omega2.id) |])
+      | Omega2.DIVIDE_AND_APPROX (e1,e2,k,d) :: l ->
+	  mkApp (Lazy.force coq_s_div_approx,
+		 [| mk_Z k; mk_Z d; 
+		    reified_of_omega env e2.Omega2.body e2.Omega2.constant;
+		    mk_nat (List.length e2.Omega2.body); 
+		    loop env_hyp l; mk_nat (get_hyp env_hyp e1.Omega2.id) |])
+      | Omega2.NOT_EXACT_DIVIDE (e1,k) :: l ->
+	  let e2_constant = Omega2.floor_div e1.Omega2.constant k in
+	  let d = e1.Omega2.constant - e2_constant * k in
+	  let e2_body = Omega2.map_eq_linear (fun c -> c / k) e1.Omega2.body in
+          mkApp (Lazy.force coq_s_not_exact_divide,
+		 [|mk_Z k; mk_Z d; 
+		   reified_of_omega env e2_body e2_constant; 
+		   mk_nat (List.length e2_body); 
+		   mk_nat (get_hyp env_hyp e1.Omega2.id)|])
+      | Omega2.EXACT_DIVIDE (e1,k) :: l ->
+	  let e2_body =  
+	    Omega2.map_eq_linear (fun c -> c / k) e1.Omega2.body in
+	  let e2_constant = Omega2.floor_div e1.Omega2.constant k in
+          mkApp (Lazy.force coq_s_exact_divide,
+		 [|mk_Z k; 
+		   reified_of_omega env e2_body e2_constant; 
+		   mk_nat (List.length e2_body);
+		   loop env_hyp l; mk_nat (get_hyp env_hyp e1.Omega2.id)|])
+      | (Omega2.MERGE_EQ(e3,e1,e2)) :: l ->
+	  let n1 = get_hyp env_hyp e1.Omega2.id and n2 = get_hyp env_hyp e2 in
+          mkApp (Lazy.force coq_s_merge_eq,
+		 [| mk_nat (List.length e1.Omega2.body);
+		    mk_nat n1; mk_nat n2;
+		    loop (CCEqua e3:: env_hyp) l |])
+      | Omega2.SUM(e3,(k1,e1),(k2,e2)) :: l ->
+	  let n1 = get_hyp env_hyp e1.Omega2.id 
+	  and n2 = get_hyp env_hyp e2.Omega2.id in
+	  let trace = shuffle_path k1 e1.Omega2.body k2 e2.Omega2.body in
+          mkApp (Lazy.force coq_s_sum,
+		 [| mk_Z k1; mk_nat n1; mk_Z k2;
+		    mk_nat n2; trace; (loop (CCEqua e3 :: env_hyp) l) |])
+      | Omega2.CONSTANT_NOT_NUL(e,k) :: l -> 
+          mkApp (Lazy.force coq_s_constant_not_nul,
+		 [|  mk_nat (get_hyp env_hyp e) |])
+      | Omega2.CONSTANT_NEG(e,k) :: l ->
+          mkApp (Lazy.force coq_s_constant_neg,
+		 [|  mk_nat (get_hyp env_hyp e) |])
+      | Omega2.STATE {Omega2.st_new_eq=new_eq; Omega2.st_def =def; 
+		      Omega2.st_orig=orig; Omega2.st_coef=m;
+                      Omega2.st_var=sigma } :: l ->
+	  let n1 = get_hyp env_hyp orig.Omega2.id 
+	  and n2 = get_hyp env_hyp def.Omega2.id in
+	  let v = unintern_omega env sigma in
+	  let o_def = oformula_of_omega env def in
+	  let o_orig = oformula_of_omega env orig in
+	  let body =
+	     Oplus (o_orig,Omult (Oplus (Oopp v,o_def), Oint m)) in
+          let trace,_ = normalize_linear_term env body in
+	  mkApp (Lazy.force coq_s_state,
+		 [| mk_Z m; trace; mk_nat n1; mk_nat n2; 
+		    loop (CCEqua new_eq.Omega2.id :: env_hyp) l |])
+      |	Omega2.HYP _ :: l -> loop env_hyp l
+      |	Omega2.CONSTANT_NUL e :: l ->
+	  mkApp (Lazy.force coq_s_constant_nul, 
+		 [|  mk_nat (get_hyp env_hyp e) |])
+      |	Omega2.NEGATE_CONTRADICT(e1,e2,b) :: l ->
+	  mkApp (Lazy.force coq_s_negate_contradict, 
+		 [|  mk_nat (get_hyp env_hyp e1.Omega2.id);
+		     mk_nat (get_hyp env_hyp e2.Omega2.id) |])
+      | Omega2.SPLIT_INEQ(e,(e1,l1),(e2,l2)) :: l ->
+	  let i =  get_hyp env_hyp e.Omega2.id in
+	  let r1 = loop (CCEqua e1 :: env_hyp) l1 in
+	  let r2 = loop (CCEqua e2 :: env_hyp) l2 in
+	  mkApp (Lazy.force coq_s_split_ineq, 
+                  [| mk_nat (List.length e.Omega2.body); mk_nat i; r1 ; r2 |])
+      |	(Omega2.FORGET_C _ | Omega2.FORGET _ | Omega2.FORGET_I _) :: l ->
+	  loop env_hyp l
+      | (Omega2.WEAKEN  _ ) :: l -> failwith "not_treated"
+      |	[] -> failwith "no contradiction"
+  in loop env_hyp
+
+let rec decompose_tree env ctxt = function
+   Tree(i,left,right) ->
+     let org = 
+       try Hashtbl.find env.constructors i 
+       with Not_found ->
+	 failwith (Printf.sprintf "Cannot find constructor %d" i) in
+     let (index,path) = find_path org ctxt in
+     let left_hyp = CCHyp{o_hyp=org.o_hyp;o_path=org.o_path @ [O_left]} in
+     let right_hyp = CCHyp{o_hyp=org.o_hyp;o_path=org.o_path @ [O_right]} in
+     app coq_e_split 
+       [| mk_nat index;
+	  mk_direction_list path;
+	  decompose_tree env (left_hyp::ctxt) left;
+	  decompose_tree env (right_hyp::ctxt) right |]
+  | Leaf s ->
+      decompose_tree_hyps s.s_trace env ctxt  s.s_equa_deps
+and decompose_tree_hyps trace env ctxt = function
+    [] -> app coq_e_solve [| replay_history env ctxt trace |]
+  | (i::l) -> 
+      let equation =
+        try Hashtbl.find env.equations i 
+        with Not_found ->
+	  failwith (Printf.sprintf "Cannot find equation %d" i) in
+      let (index,path) = find_path equation.e_origin ctxt in
+      let full_path = if equation.e_negated then path @ [O_mono] else path in
+      let cont = 
+	decompose_tree_hyps trace env 
+	   (CCEqua equation.e_omega.Omega2.id :: ctxt) l in
+      app coq_e_extract [|mk_nat index;
+			  mk_direction_list full_path;
+			  cont |]
+
+(* \section{La fonction principale} *)
+ (* Cette fonction construit la
+trace pour la procédure de décision réflexive.  A partir des résultats
+de l'extraction des systèmes, elle lance la résolution par Omega, puis
+l'extraction d'un ensemble minimal de solutions permettant la
+résolution globale du système et enfin construit la trace qui permet
+de faire rejouer cette solution par la tactique réflexive. *)
+
+let resolution env full_reified_goal systems_list =
+  let num = ref 0 in
+  let solve_system list_eq = 
+    let index = !num in
+    let system = List.map (fun eq -> eq.e_omega) list_eq in
+    let trace = 
+      Omega2.simplify_strong 
+	((fun () -> new_eq_id env),new_omega_id,display_omega_id) 
+	system in 
+    (* calcule les hypotheses utilisées pour la solution *)
+    let vars = hyps_used_in_trace trace in
+    let splits = get_eclatement env vars in
+    if !debug then begin
+      Printf.printf "SYSTEME %d\n" index;
+      Omega2.display_action display_omega_id trace;
+      print_string "\n  Depend :";
+      List.iter (fun i -> Printf.printf " %d" i) vars;
+      print_string "\n  Split points :";
+      List.iter display_depend splits;
+      Printf.printf "\n------------------------------------\n"
+    end;
+    incr num;
+    {s_index = index; s_trace = trace; s_equa_deps = vars}, splits  in
+  if !debug then Printf.printf "\n====================================\n";
+  let all_solutions = List.map solve_system systems_list in
+  let solution_tree = solve_with_constraints all_solutions [] in
+  if !debug then  begin
+    display_solution_tree stdout solution_tree;
+    print_newline()
+  end;
+  (* calcule la liste de toutes les hypothèses utilisées dans l'arbre de solution *)
+  let useful_equa_id = list_uniq (equas_of_solution_tree solution_tree) in
+  (* recupere explicitement ces equations *)
+  let equations = List.map (get_equation env) useful_equa_id in
+  let l_hyps' = list_uniq (List.map (fun e -> e.e_origin.o_hyp) equations) in
+  let l_hyps = id_concl :: list_remove id_concl l_hyps' in
+  let useful_hyps =
+    List.map (fun id -> List.assoc id full_reified_goal) l_hyps in
+  let useful_vars = vars_of_equations equations in
+  (* variables a introduire *)
+  let to_introduce = add_stated_equations env solution_tree in
+  let stated_vars =  List.map (fun (v,_,_,_) -> v) to_introduce in
+  let l_generalize_arg = List.map (fun (_,t,_,_) -> t) to_introduce in
+  let hyp_stated_vars = List.map (fun (_,_,_,id) -> CCEqua id) to_introduce in
+  (* L'environnement de base se construit en deux morceaux :
+     - les variables des équations utiles
+     - les nouvelles variables declarées durant les preuves *)
+  let all_vars_env = useful_vars @ stated_vars in
+  let basic_env =
+    let rec loop i = function
+	var :: l -> 
+	  let t = get_reified_atom env var in 
+	  Hashtbl.add env.real_indices var i; t :: loop (i+1) l
+      |	[] -> [] in
+    loop 0 all_vars_env in
+  let env_terms_reified = mk_list (Lazy.force coq_Z) basic_env in
+  (* On peut maintenant généraliser le but : env est a jour *)
+  let l_reified_stated =
+     List.map (fun (_,_,(l,r),_) -> 
+                 app coq_p_eq [| reified_of_formula env l; 
+                                 reified_of_formula env r |])
+              to_introduce in
+  let reified_concl = 
+    match useful_hyps with
+      (Pnot p) :: _ ->
+	reified_of_proposition env (really_useful_prop useful_equa_id p)
+    | _ ->  reified_of_proposition env Pfalse in
+  let l_reified_terms =
+    (List.map
+      (fun p ->
+         reified_of_proposition env (really_useful_prop useful_equa_id p))
+      (List.tl useful_hyps)) in
+  let env_props_reified = mk_plist env.props in
+  let reified_goal = 
+    mk_list (Lazy.force coq_proposition)
+            (l_reified_stated @ l_reified_terms) in
+  let reified = 
+    app coq_interp_sequent 
+       [| env_props_reified;env_terms_reified;reified_concl;reified_goal |] in
+  let normalize_equation e = 
+    let rec loop = function
+	[] -> app (if e.e_negated then coq_p_invert else coq_p_step)
+	          [| e.e_trace |]
+      |	((O_left | O_mono) :: l) -> app coq_p_left [| loop l |]
+      |	(O_right :: l) -> app coq_p_right [| loop l |] in
+    app coq_pair_step 
+        [| mk_nat (list_index e.e_origin.o_hyp l_hyps) ;
+	   loop e.e_origin.o_path |] in
+  let normalization_trace =
+    mk_list (Lazy.force coq_h_step) (List.map normalize_equation equations) in
+
+  let initial_context =
+    List.map (fun id -> CCHyp{o_hyp=id;o_path=[]}) (List.tl l_hyps) in
+  let context = 
+    CCHyp{o_hyp=id_concl;o_path=[]} :: hyp_stated_vars @ initial_context in
+  let decompose_tactic = decompose_tree env context solution_tree in
+
+  Tactics.generalize 
+    (l_generalize_arg @ List.map Term.mkVar (List.tl l_hyps)) >>
+  Tactics.change_in_concl None reified >> 
+  Tactics.apply (app coq_do_omega [|decompose_tactic; normalization_trace|]) >>
+  show_goal >>
+  Tactics.normalise_in_concl >>
+  Tactics.apply (Lazy.force coq_I)
+
+let total_reflexive_omega_tactic gl = 
+  if !Options.v7 then Util.error "ROmega does not work in v7 mode";
+  try
+  let env = new_environment () in
+  let full_reified_goal = reify_gl env gl in
+  let systems_list = destructurate_hyps full_reified_goal in
+  if !debug then begin 
+    display_systems systems_list
+  end;
+  resolution env full_reified_goal systems_list gl
+  with Omega2.NO_CONTRADICTION -> Util.error "ROmega can't solve this system"
+
+
+(*i let tester = Tacmach.hide_atomic_tactic "TestOmega" test_tactic i*)
+
+