Imported Upstream snapshot 8.3~beta0+13298

7 files changed, 0 insertions, 5101 deletions

diff --git a/contrib/romega/README b/contrib/romega/README
deleted file mode 100644
index 86c9e58a..00000000
--- a/contrib/romega/README
+++ /dev/null
@@ -1,6 +0,0 @@
-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
deleted file mode 100644
index 4281cc57..00000000
--- a/contrib/romega/ROmega.v
+++ /dev/null
@@ -1,12 +0,0 @@
-(*************************************************************************
-
-   PROJET RNRT Calife - 2001
-   Author: Pierre Crégut - France Télécom R&D
-   Licence : LGPL version 2.1
-
- *************************************************************************)
-
-Require Import ReflOmegaCore.
-Require Export Setoid.
-Require Export PreOmega.
-Require Export ZArith_base.
diff --git a/contrib/romega/ReflOmegaCore.v b/contrib/romega/ReflOmegaCore.v
deleted file mode 100644
index 12176d66..00000000
--- a/contrib/romega/ReflOmegaCore.v
+++ /dev/null
@@ -1,3216 +0,0 @@
-(* -*- coding: utf-8 -*- *)
-(*************************************************************************
-
-   PROJET RNRT Calife - 2001
-   Author: Pierre Crégut - France Télécom R&D
-   Licence du projet : LGPL version 2.1
-
- *************************************************************************)
-
-Require Import List Bool Sumbool EqNat Setoid Ring_theory Decidable ZArith_base.
-Delimit Scope Int_scope with I.
-
-(* Abstract Integers. *)
-
-Module Type Int. 
-
-  Parameter int : Set. 
-
-  Parameter zero : int. 
-  Parameter one : int. 
-  Parameter plus : int -> int -> int. 
-  Parameter opp : int -> int.
-  Parameter minus : int -> int -> int. 
-  Parameter mult : int -> int -> int.
-
-  Notation "0" := zero : Int_scope.
-  Notation "1" := one : Int_scope. 
-  Infix "+" := plus : Int_scope.
-  Infix "-" := minus : Int_scope.
-  Infix "*" := mult : Int_scope.
-  Notation "- x" := (opp x) : Int_scope.
-
-  Open Scope Int_scope.
-
-  (* First, int is a ring: *)
-  Axiom ring : @ring_theory int 0 1 plus mult minus opp (@eq int).
-
-  (* int should also be ordered: *)
-
-  Parameter le : int -> int -> Prop.
-  Parameter lt : int -> int -> Prop.
-  Parameter ge : int -> int -> Prop.
-  Parameter gt : int -> int -> Prop.
-  Notation "x <= y" := (le x y): Int_scope.
-  Notation "x < y" := (lt x y) : Int_scope.
-  Notation "x >= y" := (ge x y) : Int_scope.
-  Notation "x > y" := (gt x y): Int_scope.
-  Axiom le_lt_iff : forall i j, (i<=j) <-> ~(j<i).
-  Axiom ge_le_iff : forall i j, (i>=j) <-> (j<=i).
-  Axiom gt_lt_iff : forall i j, (i>j) <-> (j<i).
-
-  (* Basic properties of this order *)
-  Axiom lt_trans : forall i j k, i<j -> j<k -> i<k.
-  Axiom lt_not_eq : forall i j, i<j -> i<>j.
-
-  (* Compatibilities *)
-  Axiom lt_0_1 : 0<1.
-  Axiom plus_le_compat : forall i j k l, i<=j -> k<=l -> i+k<=j+l.
-  Axiom opp_le_compat : forall i j, i<=j -> (-j)<=(-i).
-  Axiom mult_lt_compat_l : 
-   forall i j k, 0 < k -> i < j -> k*i<k*j.
-
-  (* We should have a way to decide the equality and the order*)  
-  Parameter compare : int -> int -> comparison.
-  Infix "?=" := compare (at level 70, no associativity) : Int_scope.
-  Axiom compare_Eq : forall i j, compare i j = Eq <-> i=j.
-  Axiom compare_Lt : forall i j, compare i j = Lt <-> i<j.
-  Axiom compare_Gt : forall i j, compare i j = Gt <-> i>j.
-
-  (* Up to here, these requirements could be fulfilled 
-     by any totally ordered ring. Let's now be int-specific: *)
-  Axiom le_lt_int : forall x y, x<y <-> x<=y+-(1).
-
-  (* Btw, lt_0_1 could be deduced from this last axiom *)
-
-End Int.
-
-
-
-(* Of course, Z is a model for our abstract int *)
-
-Module Z_as_Int <: Int.
-
-  Open Scope Z_scope.
-
-  Definition int := Z. 
-  Definition zero := 0. 
-  Definition one := 1. 
-  Definition plus := Zplus.
-  Definition opp := Zopp.
-  Definition minus := Zminus.
-  Definition mult := Zmult.
-
-  Lemma ring : @ring_theory int zero one plus mult minus opp (@eq int).
-  Proof.
-  constructor.
-  exact Zplus_0_l.
-  exact Zplus_comm.
-  exact Zplus_assoc.
-  exact Zmult_1_l.
-  exact Zmult_comm.
-  exact Zmult_assoc.
-  exact Zmult_plus_distr_l.
-  unfold minus, Zminus; auto.
-  exact Zplus_opp_r.
-  Qed.
-
-  Definition le := Zle.
-  Definition lt := Zlt.
-  Definition ge := Zge.
-  Definition gt := Zgt.
-  Lemma le_lt_iff : forall i j, (i<=j) <-> ~(j<i).
-  Proof.
-   split; intros.
-   apply Zle_not_lt; auto.
-   rewrite <- Zge_iff_le.
-   apply Znot_lt_ge; auto.
-  Qed.
-  Definition ge_le_iff := Zge_iff_le.
-  Definition gt_lt_iff := Zgt_iff_lt.
-
-  Definition lt_trans := Zlt_trans.
-  Definition lt_not_eq := Zlt_not_eq.
-
-  Definition lt_0_1 := Zlt_0_1.
-  Definition plus_le_compat := Zplus_le_compat.
-  Definition mult_lt_compat_l := Zmult_lt_compat_l.
-  Lemma opp_le_compat : forall i j, i<=j -> (-j)<=(-i).
-  Proof.
-  unfold Zle; intros; rewrite <- Zcompare_opp; auto.
-  Qed.
-
-  Definition compare := Zcompare.
-  Definition compare_Eq := Zcompare_Eq_iff_eq.
-  Lemma compare_Lt : forall i j, compare i j = Lt <-> i<j.
-  Proof. intros; unfold compare, Zlt; intuition. Qed.
-  Lemma compare_Gt : forall i j, compare i j = Gt <-> i>j.
-  Proof. intros; unfold compare, Zgt; intuition. Qed.
-
-  Lemma le_lt_int : forall x y, x<y <-> x<=y+-(1).
-  Proof.
-   intros; split; intros.
-   generalize (Zlt_left _ _ H); simpl; intros.
-   apply Zle_left_rev; auto.
-   apply Zlt_0_minus_lt.
-   generalize (Zplus_le_lt_compat x (y+-1) (-x) (-x+1) H).
-   rewrite Zplus_opp_r.
-   rewrite <-Zplus_assoc.
-   rewrite (Zplus_permute (-1)).
-   simpl in *.
-   rewrite Zplus_0_r.
-   intro H'; apply H'.
-   replace (-x+1) with (Zsucc (-x)); auto.
-   apply Zlt_succ.
-  Qed.
-
-End Z_as_Int.  
-
-
-
-
-Module IntProperties (I:Int). 
- Import I.
- 
- (* Primo, some consequences of being a ring theory... *)
- 
- Definition two := 1+1.
- Notation "2" := two : Int_scope. 
-
- (* Aliases for properties packed in the ring record. *)
-
- Definition plus_assoc := ring.(Radd_assoc).
- Definition plus_comm := ring.(Radd_comm).
- Definition plus_0_l := ring.(Radd_0_l).
- Definition mult_assoc := ring.(Rmul_assoc). 
- Definition mult_comm := ring.(Rmul_comm).
- Definition mult_1_l := ring.(Rmul_1_l).
- Definition mult_plus_distr_r := ring.(Rdistr_l).
- Definition opp_def := ring.(Ropp_def).
- Definition minus_def := ring.(Rsub_def).
-
- Opaque plus_assoc plus_comm plus_0_l mult_assoc mult_comm mult_1_l 
-  mult_plus_distr_r opp_def minus_def.
-
- (* More facts about plus *)
-
- Lemma plus_0_r : forall x, x+0 = x.
- Proof. intros; rewrite plus_comm; apply plus_0_l. Qed.
-
- Lemma plus_0_r_reverse : forall x, x = x+0.
- Proof. intros; symmetry; apply plus_0_r. Qed. 
-
- Lemma plus_assoc_reverse : forall x y z, x+y+z = x+(y+z).
- Proof. intros; symmetry; apply plus_assoc. Qed.
-
- Lemma plus_permute : forall x y z, x+(y+z) = y+(x+z).
- Proof. intros; do 2 rewrite plus_assoc; f_equal; apply plus_comm. Qed.
-
- Lemma plus_reg_l : forall x y z, x+y = x+z -> y = z.
- Proof. 
-  intros.
-  rewrite (plus_0_r_reverse y), (plus_0_r_reverse z), <-(opp_def x).
-  now rewrite plus_permute, plus_assoc, H, <- plus_assoc, plus_permute. 
- Qed.
-
- (* More facts about mult *) 
-   
- Lemma mult_assoc_reverse : forall x y z, x*y*z = x*(y*z).
- Proof. intros; symmetry; apply mult_assoc. Qed.
-
- Lemma mult_plus_distr_l : forall x y z, x*(y+z)=x*y+x*z.
- Proof.
-  intros.
-  rewrite (mult_comm x (y+z)), (mult_comm x y), (mult_comm x z).
-  apply mult_plus_distr_r.
- Qed.
-
- Lemma mult_0_l : forall x, 0*x = 0.
- Proof. 
-  intros.
-  generalize (mult_plus_distr_r 0 1 x).
-  rewrite plus_0_l, mult_1_l, plus_comm; intros.
-  apply plus_reg_l with x.
-  rewrite <- H.
-  apply plus_0_r_reverse.
- Qed.
- 
-
- (* More facts about opp *)
-
- Definition plus_opp_r := opp_def.
-
- Lemma plus_opp_l : forall x, -x + x = 0.
- Proof. intros; now rewrite plus_comm, opp_def. Qed.
-
- Lemma mult_opp_comm : forall x y, - x * y = x * - y.
- Proof.
-  intros.
-  apply plus_reg_l with (x*y).
-  rewrite <- mult_plus_distr_l, <- mult_plus_distr_r.
-  now rewrite opp_def, opp_def, mult_0_l, mult_comm, mult_0_l.
- Qed.
-
- Lemma opp_eq_mult_neg_1 : forall x, -x = x * -(1).
- Proof.
-  intros; now rewrite mult_comm, mult_opp_comm, mult_1_l.
- Qed.
-
- Lemma opp_involutive : forall x, -(-x) = x.
- Proof.
-  intros.
-  apply plus_reg_l with (-x).
-  now rewrite opp_def, plus_comm, opp_def.
- Qed.
-
- Lemma opp_plus_distr : forall x y, -(x+y) = -x + -y.
- Proof.
-  intros.
-  apply plus_reg_l with (x+y).
-  rewrite opp_def.
-  rewrite plus_permute.
-  do 2 rewrite plus_assoc.
-  now rewrite (plus_comm (-x)), opp_def, plus_0_l, opp_def.
- Qed.
-
- Lemma opp_mult_distr_r : forall x y, -(x*y) = x * -y.
- Proof.
-  intros.
-  rewrite <- mult_opp_comm.
-  apply plus_reg_l with (x*y).
-  now rewrite opp_def, <-mult_plus_distr_r, opp_def, mult_0_l.
- Qed.  
-
- Lemma egal_left : forall n m, n=m -> n+-m = 0.
- Proof. intros; subst; apply opp_def. Qed.
-
- Lemma ne_left_2 : forall x y : int, x<>y -> 0<>(x + - y).
- Proof.
- intros; contradict H.
- apply (plus_reg_l (-y)).
- now rewrite plus_opp_l, plus_comm, H.
- Qed.
-
- (* Special lemmas for factorisation. *)
-
- Lemma red_factor0 : forall n, n = n*1.
- Proof. symmetry; rewrite mult_comm; apply mult_1_l. Qed.
-
- Lemma red_factor1 : forall n, n+n = n*2.
- Proof. 
-  intros; unfold two.
-  now rewrite mult_comm, mult_plus_distr_r, mult_1_l.
- Qed.
-
- Lemma red_factor2 : forall n m, n + n*m = n * (1+m).
- Proof.
-  intros; rewrite mult_plus_distr_l.
-  f_equal; now rewrite mult_comm, mult_1_l.
- Qed.
-
- Lemma red_factor3 : forall n m, n*m + n = n*(1+m).
- Proof. intros; now rewrite plus_comm, red_factor2. Qed.
-
- Lemma red_factor4 : forall n m p, n*m + n*p = n*(m+p).
- Proof. 
-  intros; now rewrite mult_plus_distr_l.
- Qed.
- 
- Lemma red_factor5 : forall n m , n * 0 + m = m.
- Proof. intros; now rewrite mult_comm, mult_0_l, plus_0_l. Qed.
-
- Definition red_factor6 := plus_0_r_reverse.
-
-
- (* Specialized distributivities *)
-
- Hint Rewrite mult_plus_distr_l mult_plus_distr_r mult_assoc : int.
- Hint Rewrite <- plus_assoc : int.
-
- Lemma OMEGA10 :
-  forall v c1 c2 l1 l2 k1 k2 : int,
-  (v * c1 + l1) * k1 + (v * c2 + l2) * k2 =
-  v * (c1 * k1 + c2 * k2) + (l1 * k1 + l2 * k2).
- Proof.
-  intros; autorewrite with int; f_equal; now rewrite plus_permute.
- Qed.
-
- Lemma OMEGA11 :
-  forall v1 c1 l1 l2 k1 : int,
-  (v1 * c1 + l1) * k1 + l2 = v1 * (c1 * k1) + (l1 * k1 + l2).
- Proof.
-  intros; now autorewrite with int.
- Qed.
-
- Lemma OMEGA12 :
-  forall v2 c2 l1 l2 k2 : int,
-  l1 + (v2 * c2 + l2) * k2 = v2 * (c2 * k2) + (l1 + l2 * k2).
- Proof.
-  intros; autorewrite with int; now rewrite plus_permute.
- Qed.
-
- Lemma OMEGA13 :
-  forall v l1 l2 x : int,
-  v * -x + l1 + (v * x + l2) = l1 + l2.
- Proof.
- intros; autorewrite with int.
- rewrite plus_permute; f_equal.
- rewrite plus_assoc.
- now rewrite <- mult_plus_distr_l, plus_opp_l, mult_comm, mult_0_l, plus_0_l.
- Qed.
-
- Lemma OMEGA15 :
-  forall v c1 c2 l1 l2 k2 : int,
-  v * c1 + l1 + (v * c2 + l2) * k2 = v * (c1 + c2 * k2) + (l1 + l2 * k2).
- Proof.
-  intros; autorewrite with int; f_equal; now rewrite plus_permute.
- Qed.
-
- Lemma OMEGA16 : forall v c l k : int, (v * c + l) * k = v * (c * k) + l * k.
- Proof.
-  intros; now autorewrite with int.
- Qed.
-
- Lemma sum1 : forall a b c d : int, 0 = a -> 0 = b -> 0 = a * c + b * d.
- Proof.
- intros; elim H; elim H0; simpl in |- *; auto.
- now rewrite mult_0_l, mult_0_l, plus_0_l.
- Qed.
-
-
- (* Secondo, some results about order (and equality) *)  
-
- Lemma lt_irrefl : forall n, ~ n<n.
- Proof.
- intros n H.
- elim (lt_not_eq _ _ H); auto.
- Qed.
-
- Lemma lt_antisym : forall n m, n<m -> m<n -> False.
- Proof.
- intros; elim (lt_irrefl _ (lt_trans _ _ _ H H0)); auto.
- Qed.
-
- Lemma lt_le_weak : forall n m, n<m -> n<=m.
- Proof.
-  intros; rewrite le_lt_iff; intro H'; eapply lt_antisym; eauto.
- Qed.
-
- Lemma le_refl : forall n, n<=n.
- Proof.
- intros; rewrite le_lt_iff; apply lt_irrefl; auto.
- Qed.
-
- Lemma le_antisym : forall n m, n<=m -> m<=n -> n=m.
- Proof.
- intros n m; do 2 rewrite le_lt_iff; intros.
- rewrite <- compare_Lt in H0.
- rewrite <- gt_lt_iff, <- compare_Gt in H.
- rewrite <- compare_Eq.
- destruct compare; intuition.
- Qed.
-
- Lemma lt_eq_lt_dec : forall n m, { n<m }+{ n=m }+{ m<n }.
- Proof.
-  intros.
-  generalize (compare_Lt n m)(compare_Eq n m)(compare_Gt n m).
-  destruct compare; [ left; right | left; left | right ]; intuition.
-  rewrite gt_lt_iff in H1; intuition.
- Qed.
-
- Lemma lt_dec : forall n m: int, { n<m } + { ~n<m }.
- Proof.
-  intros.
-  generalize (compare_Lt n m)(compare_Eq n m)(compare_Gt n m).
-  destruct compare; [ right | left | right ]; intuition discriminate.
- Qed.
-
- Lemma lt_le_iff : forall n m, (n<m) <-> ~(m<=n).
- Proof.
-  intros.
-  rewrite le_lt_iff.
-  destruct (lt_dec n m); intuition.
- Qed.
-
- Lemma le_dec : forall n m: int, { n<=m } + { ~n<=m }.
- Proof.
-  intros; destruct (lt_dec m n); [right|left]; rewrite le_lt_iff; intuition.
- Qed.
-
- Lemma le_lt_dec : forall n m, { n<=m } + { m<n }.
- Proof.
-  intros; destruct (le_dec n m); [left|right]; auto; now rewrite lt_le_iff.
- Qed.
-
-
- Definition beq i j := match compare i j with Eq => true | _ => false end.
-
- Lemma beq_iff : forall i j, beq i j = true <-> i=j.
- Proof.
- intros; unfold beq; generalize (compare_Eq i j).
- destruct compare; intuition discriminate.
- Qed.
-                               
- Lemma beq_true : forall i j, beq i j = true -> i=j.
- Proof.
- intros.
- rewrite <- beq_iff; auto.
- Qed.
-
- Lemma beq_false : forall i j, beq i j = false -> i<>j.
- Proof.
- intros.
- intro H'.
- rewrite <- beq_iff in H'; rewrite H' in H; discriminate.
- Qed.
-
- Lemma eq_dec : forall n m:int, { n=m } + { n<>m }.
- Proof.
-  intros; generalize (beq_iff n m); destruct beq; [left|right]; intuition.
- Qed.
-
- Definition bgt i j := match compare i j with Gt => true | _ => false end.
-
- Lemma bgt_iff : forall i j, bgt i j = true <-> i>j.
- Proof.
- intros; unfold bgt; generalize (compare_Gt i j).
- destruct compare; intuition discriminate.
- Qed.
-
- Lemma bgt_true : forall i j, bgt i j = true -> i>j.
- Proof. intros; now rewrite <- bgt_iff. Qed.
-
- Lemma bgt_false : forall i j, bgt i j = false -> i<=j.
- Proof. 
-  intros.
-  rewrite le_lt_iff, <-gt_lt_iff, <-bgt_iff; intro H'; now rewrite H' in H.
- Qed.
-
- Lemma le_is_lt_or_eq : forall n m, n<=m -> { n<m } + { n=m }.
- Proof.
-  intros.
-  destruct (eq_dec n m) as [H'|H'].
-  right; intuition.
-  left; rewrite lt_le_iff.
-  contradict H'.
-  apply le_antisym; auto.
- Qed.
-
- Lemma le_neq_lt : forall n m, n<=m -> n<>m -> n<m.
- Proof.
-  intros.
-  destruct (le_is_lt_or_eq _ _ H); intuition.
- Qed.
-
- Lemma le_trans : forall n m p, n<=m -> m<=p -> n<=p.
- Proof.
-  intros n m p; do 3 rewrite le_lt_iff; intros A B C.
-  destruct (lt_eq_lt_dec p m) as [[H|H]|H]; subst; auto.
-  generalize (lt_trans _ _ _ H C); intuition.
- Qed.
- 
- (* order and operations *)
-
- Lemma le_0_neg : forall n, 0 <= n <-> -n <= 0.
- Proof.
- intros.
- pattern 0 at 2; rewrite <- (mult_0_l (-(1))).
- rewrite <- opp_eq_mult_neg_1.
- split; intros.
- apply opp_le_compat; auto.
- rewrite <-(opp_involutive 0), <-(opp_involutive n).
- apply opp_le_compat; auto.
- Qed.
-
- Lemma le_0_neg' : forall n, n <= 0 <-> 0 <= -n.
- Proof.
- intros; rewrite le_0_neg, opp_involutive; intuition.
- Qed.
-
- Lemma plus_le_reg_r : forall n m p, n + p <= m + p -> n <= m.
- Proof.
- intros.
- replace n with ((n+p)+-p).
- replace m with ((m+p)+-p).
- apply plus_le_compat; auto.
- apply le_refl.
- now rewrite <- plus_assoc, opp_def, plus_0_r.
- now rewrite <- plus_assoc, opp_def, plus_0_r.
- Qed.
-
- Lemma plus_le_lt_compat : forall n m p q, n<=m -> p<q -> n+p<m+q.
- Proof.
- intros.
- apply le_neq_lt.
- apply plus_le_compat; auto.
- apply lt_le_weak; auto.
- rewrite lt_le_iff in H0.
- contradict H0.
- apply plus_le_reg_r with m.
- rewrite (plus_comm q m), <-H0, (plus_comm p m).
- apply plus_le_compat; auto.
- apply le_refl; auto.
- Qed.
-
- Lemma plus_lt_compat : forall n m p q, n<m -> p<q -> n+p<m+q.
- Proof.
- intros.
- apply plus_le_lt_compat; auto.
- apply lt_le_weak; auto.
- Qed.
-
- Lemma opp_lt_compat : forall n m, n<m -> -m < -n.
- Proof.
- intros n m; do 2 rewrite lt_le_iff; intros H; contradict H.
- rewrite <-(opp_involutive m), <-(opp_involutive n).
- apply opp_le_compat; auto.
- Qed.
-
- Lemma lt_0_neg : forall n, 0 < n <-> -n < 0.
- Proof.
- intros.
- pattern 0 at 2; rewrite <- (mult_0_l (-(1))).
- rewrite <- opp_eq_mult_neg_1.
- split; intros.
- apply opp_lt_compat; auto.
- rewrite <-(opp_involutive 0), <-(opp_involutive n).
- apply opp_lt_compat; auto.
- Qed.
-
- Lemma lt_0_neg' : forall n, n < 0 <-> 0 < -n.
- Proof.
- intros; rewrite lt_0_neg, opp_involutive; intuition.
- Qed.
-
- Lemma mult_lt_0_compat : forall n m, 0 < n -> 0 < m -> 0 < n*m.
- Proof.
- intros.
- rewrite <- (mult_0_l n), mult_comm.
- apply mult_lt_compat_l; auto.
- Qed.
-
- Lemma mult_integral : forall n m, n * m = 0 -> n = 0 \/ m = 0.
- Proof.
- intros.
- destruct (lt_eq_lt_dec n 0) as [[Hn|Hn]|Hn]; auto; 
-  destruct (lt_eq_lt_dec m 0) as [[Hm|Hm]|Hm]; auto; elimtype False.
-
- rewrite lt_0_neg' in Hn.
- rewrite lt_0_neg' in Hm.
- generalize (mult_lt_0_compat _ _ Hn Hm).
- rewrite <- opp_mult_distr_r, mult_comm, <- opp_mult_distr_r, opp_involutive.
- rewrite mult_comm, H.
- exact (lt_irrefl 0).
-
- rewrite lt_0_neg' in Hn.
- generalize (mult_lt_0_compat _ _ Hn Hm).
- rewrite mult_comm, <- opp_mult_distr_r, mult_comm.
- rewrite H.
- rewrite opp_eq_mult_neg_1, mult_0_l.
- exact (lt_irrefl 0).
-
- rewrite lt_0_neg' in Hm.
- generalize (mult_lt_0_compat _ _ Hn Hm).
- rewrite <- opp_mult_distr_r.
- rewrite H.
- rewrite opp_eq_mult_neg_1, mult_0_l.
- exact (lt_irrefl 0).
-
- generalize (mult_lt_0_compat _ _ Hn Hm).
- rewrite H.
- exact (lt_irrefl 0).
- Qed.
-
- Lemma mult_le_compat : 
-   forall i j k l, i<=j -> k<=l -> 0<=i -> 0<=k -> i*k<=j*l.
- Proof.
- intros.
- destruct (le_is_lt_or_eq _ _ H1).
-
- apply le_trans with (i*l).
- destruct (le_is_lt_or_eq _ _ H0); [ | subst; apply le_refl].
- apply lt_le_weak.
- apply mult_lt_compat_l; auto.
-
- generalize (le_trans _ _ _ H2 H0); clear H0 H1 H2; intros.
- rewrite (mult_comm i), (mult_comm j).
- destruct (le_is_lt_or_eq _ _ H0); 
-  [ | subst; do 2 rewrite mult_0_l; apply le_refl].
- destruct (le_is_lt_or_eq _ _ H); 
-  [ | subst; apply le_refl].
- apply lt_le_weak.
- apply mult_lt_compat_l; auto.
-
- subst i.
- rewrite mult_0_l.
- generalize (le_trans _ _ _ H2 H0); clear H0 H1 H2; intros.
- destruct (le_is_lt_or_eq _ _ H); 
-  [ | subst; rewrite mult_0_l; apply le_refl].
- destruct (le_is_lt_or_eq _ _ H0); 
-  [ | subst; rewrite mult_comm, mult_0_l; apply le_refl].
- apply lt_le_weak.
- apply mult_lt_0_compat; auto.
- Qed.
-
- Lemma sum5 :
- forall a b c d : int, c <> 0 -> 0 <> a -> 0 = b -> 0 <> a * c + b * d.
- Proof.
- intros.
- subst b; rewrite mult_0_l, plus_0_r.
- contradict H.
- symmetry in H; destruct (mult_integral _ _ H); congruence.
- Qed.
-
- Lemma one_neq_zero : 1 <> 0.
- Proof.
- red; intro.
- symmetry in H.
- apply (lt_not_eq 0 1); auto.
- apply lt_0_1.
- Qed.
-
- Lemma minus_one_neq_zero : -(1) <> 0.
- Proof.
- apply lt_not_eq.
- rewrite <- lt_0_neg.
- apply lt_0_1.
- Qed.
-
- Lemma le_left : forall n m, n <= m -> 0 <= m + - n.
- Proof.
-  intros.
-  rewrite <- (opp_def m).
-  apply plus_le_compat.
-  apply le_refl.
-  apply opp_le_compat; auto.
- Qed.
-
- Lemma OMEGA2 : forall x y, 0 <= x -> 0 <= y -> 0 <= x + y.
- Proof.
- intros.
- replace 0 with (0+0).
- apply plus_le_compat; auto.
- rewrite plus_0_l; auto.
- Qed.
-
- Lemma OMEGA8 : forall x y, 0 <= x -> 0 <= y -> x = - y -> x = 0.
- Proof.
- intros.
- assert (y=-x).
-  subst x; symmetry; apply opp_involutive.
- clear H1; subst y.
- destruct (eq_dec 0 x) as [H'|H']; auto.
- assert (H'':=le_neq_lt _ _ H H').
- generalize (plus_le_lt_compat _ _ _ _ H0 H'').
- rewrite plus_opp_l, plus_0_l.
- intros.
- elim (lt_not_eq _ _ H1); auto.
- Qed.
-
- Lemma sum2 :
- forall a b c d : int, 0 <= d -> 0 = a -> 0 <= b -> 0 <= a * c + b * d.
- Proof.
- intros.
- subst a; rewrite mult_0_l, plus_0_l.
- rewrite <- (mult_0_l 0).
- apply mult_le_compat; auto; apply le_refl.
- Qed.
-
- Lemma sum3 :
- forall a b c d : int,
-  0 <= c -> 0 <= d -> 0 <= a -> 0 <= b -> 0 <= a * c + b * d.
- Proof.
- intros.
- rewrite <- (plus_0_l 0).
- apply plus_le_compat; auto.
- rewrite <- (mult_0_l 0).
- apply mult_le_compat; auto; apply le_refl.
- rewrite <- (mult_0_l 0).
- apply mult_le_compat; auto; apply le_refl.
- Qed.
-
- Lemma sum4 : forall k : int, k>0 -> 0 <= k.
- Proof.
- intros k; rewrite gt_lt_iff; apply lt_le_weak.
- Qed.
-
- (* Lemmas specific to integers (they use lt_le_int) *)
-
- Lemma lt_left : forall n m, n < m -> 0 <= m + -(1) + - n.
- Proof.
- intros; apply le_left.
- now rewrite <- le_lt_int.
- Qed.
-
- Lemma lt_left_inv : forall x y, 0 <= y + -(1) + - x -> x < y.
- Proof.
- intros.
- generalize (plus_le_compat _ _ _ _ H (le_refl x)); clear H.
- now rewrite plus_0_l, <-plus_assoc, plus_opp_l, plus_0_r, le_lt_int.
- Qed.
-
- Lemma OMEGA4 : forall x y z, x > 0 -> y > x -> z * y + x <> 0.
- Proof.
- intros.
- intro H'.
- rewrite gt_lt_iff in H,H0.
- destruct (lt_eq_lt_dec z 0) as [[G|G]|G].
-
- rewrite lt_0_neg' in G.
- generalize (plus_le_lt_compat _ _ _ _ (le_refl (z*y)) H0).
- rewrite H'.
- pattern y at 2; rewrite <-(mult_1_l y), <-mult_plus_distr_r.
- intros.
- rewrite le_lt_int in G.
- rewrite <- opp_plus_distr in G.
- assert (0 < y) by (apply lt_trans with x; auto).
- generalize (mult_le_compat _ _ _ _ G (lt_le_weak _ _ H2) (le_refl 0) (le_refl 0)).
- rewrite mult_0_l, mult_comm, <- opp_mult_distr_r, mult_comm, <-le_0_neg', le_lt_iff.
- intuition.
-
- subst; rewrite mult_0_l, plus_0_l in H'; subst.
- apply (lt_not_eq _ _ H); auto.
-
- apply (lt_not_eq 0 (z*y+x)); auto.
- rewrite <- (plus_0_l 0).
- apply plus_lt_compat; auto.
- apply mult_lt_0_compat; auto.
- apply lt_trans with x; auto.
- Qed. 
-
- Lemma OMEGA19 : forall x, x<>0 -> 0 <= x + -(1) \/ 0 <= x * -(1) + -(1).
- Proof.
- intros.
- do 2 rewrite <- le_lt_int.
- rewrite <- opp_eq_mult_neg_1.
- destruct (lt_eq_lt_dec 0 x) as [[H'|H']|H'].
- auto.
- congruence.
- right.
- rewrite <-(mult_0_l (-(1))), <-(opp_eq_mult_neg_1 0).
- apply opp_lt_compat; auto.
- Qed.
-
- Lemma mult_le_approx : 
-  forall n m p, n > 0 -> n > p -> 0 <= m * n + p -> 0 <= m.
- Proof.
- intros n m p.
- do 2 rewrite gt_lt_iff.
- do 2 rewrite le_lt_iff; intros.
- contradict H1.
- rewrite lt_0_neg' in H1.
- rewrite lt_0_neg'.
- rewrite opp_plus_distr.
- rewrite mult_comm, opp_mult_distr_r.
- rewrite le_lt_int.
- rewrite <- plus_assoc, (plus_comm (-p)), plus_assoc.
- apply lt_left.
- rewrite le_lt_int.
- rewrite le_lt_int in H0.
- apply le_trans with (n+-(1)); auto.
- apply plus_le_compat; [ | apply le_refl ].
- rewrite le_lt_int in H1.
- generalize (mult_le_compat _ _ _ _ (lt_le_weak _ _ H) H1 (le_refl 0) (le_refl 0)).
- rewrite mult_0_l.
- rewrite mult_plus_distr_l.
- rewrite <- opp_eq_mult_neg_1.
- intros.
- generalize (plus_le_compat _ _ _ _ (le_refl n) H2).
- now rewrite plus_permute, opp_def, plus_0_r, plus_0_r.
- Qed.
-
- (* Some decidabilities *)
-
- Lemma dec_eq : forall i j:int, decidable (i=j).
- Proof.
-  red; intros; destruct (eq_dec i j); auto.
- Qed.
-
- Lemma dec_ne : forall i j:int, decidable (i<>j).
- Proof.
-  red; intros; destruct (eq_dec i j); auto.
- Qed.
-
- Lemma dec_le : forall i j:int, decidable (i<=j).
- Proof.
-  red; intros; destruct (le_dec i j); auto.
- Qed.
-
- Lemma dec_lt : forall i j:int, decidable (i<j).
- Proof.
-  red; intros; destruct (lt_dec i j); auto.
- Qed.
-
- Lemma dec_ge : forall i j:int, decidable (i>=j).
- Proof.
-  red; intros; rewrite ge_le_iff; destruct (le_dec j i); auto.
- Qed.
-
- Lemma dec_gt : forall i j:int, decidable (i>j).
- Proof.
-  red; intros; rewrite gt_lt_iff; destruct (lt_dec j i); auto.
- Qed.
-
-End IntProperties.
-
-
-
-
-Module IntOmega (I:Int).
-Import I.
-Module IP:=IntProperties(I).
-Import IP.
-   
-(* \subsubsection{Definition 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 : int -> term
-  | Tplus : term -> term -> term
-  | Tmult : term -> term -> term
-  | Tminus : term -> term -> term
-  | Topp : term -> term
-  | Tvar : nat -> term.
-
-Delimit Scope romega_scope with term.
-Arguments Scope Tint [Int_scope].
-Arguments Scope Tplus [romega_scope romega_scope].
-Arguments Scope Tmult [romega_scope romega_scope].
-Arguments Scope Tminus [romega_scope romega_scope].
-Arguments Scope Topp [romega_scope romega_scope].
-
-Infix "+" := Tplus : romega_scope.
-Infix "*" := Tmult : romega_scope.
-Infix "-" := Tminus : romega_scope.
-Notation "- x" := (Topp x) : romega_scope.
-Notation "[ x ]" := (Tvar x) (at level 0) : romega_scope.
-
-(* \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 (* equality between terms *)
-  | LeqTerm : term -> term -> proposition (* less or equal on terms *)
-  | TrueTerm : proposition (* true *)
-  | FalseTerm : proposition (* false *)
-  | Tnot : proposition -> proposition (* negation *)
-  | 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).      
-
-(* Definition of lists of subgoals  (set of open goals) *)
-Notation lhyps := (list hyps).
-
-(* a single goal packed in a subgoal list *)
-Notation singleton := (fun a : hyps => a :: nil).
-
-(* 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_COMM : 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_COMM : 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 and n!= 0 *)
-  | O_CONSTANT_NOT_NUL : nat -> t_omega
-  | O_CONSTANT_NEG : nat -> t_omega
-  (* division and approximation of an equation *)
-  | O_DIV_APPROX : int -> int -> term -> nat -> t_omega -> nat -> t_omega
-  (* no solution because no exact division *)
-  | O_NOT_EXACT_DIVIDE : int -> int -> term -> nat -> nat -> t_omega
-  (* exact division *)
-  | O_EXACT_DIVIDE : int -> term -> nat -> t_omega -> nat -> t_omega
-  | O_SUM : int -> nat -> int -> 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 : int -> step -> nat -> nat -> t_omega -> t_omega.
-
-(* \subsubsection{Rules for normalizing the hypothesis} *)
-(* These rules indicate how to normalize useful propositions 
-   of each useful hypothesis before the decomposition of hypothesis. 
-   The rules include the inversion phase for negation removal. *)
-
-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.
-
-(* List of normalizations to perform : with a constructor of type 
-   [p_step] allowing to visit both left and right branches, we would be  
-   able to restrict to only one normalization by hypothesis. 
-   And since all hypothesis are useful (otherwise they wouldn't be included), 
-   we would be able to replace [h_step] by a simple list. *)
-
-Inductive h_step : Set :=
-    pair_step : nat -> p_step -> h_step.
-
-(* \subsubsection{Rules for decomposing the hypothesis} *)
-(* This type allows to navigate in the logical constructors that 
-   form the predicats of the hypothesis in order to decompose them. 
-   This allows in particular to extract one hypothesis from a 
-   conjonction with possibly the right level of negations. *)
-
-Inductive direction : Set :=
-  | D_left : direction
-  | D_right : direction
-  | D_mono : direction.
-
-(* This type allows to extract useful components from hypothesis, either
-   hypothesis generated by splitting a disjonction, or equations. 
-   The last constructor indicates how to solve the obtained system 
-   via the use of the trace type of 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{Efficient decidable equality} *)
-(* For each reified data-type, we define an efficient equality test. 
-   It is not the one produced by [Decide Equality].
-   
-   Then we prove two theorem allowing to eliminate such equalities : 
-   \begin{verbatim}
-   (t1,t2: typ) (eq_typ t1 t2) = true -> t1 = t2.
-   (t1,t2: typ) (eq_typ t1 t2) = false -> ~ t1 = t2.
-   \end{verbatim} *)
-
-(* \subsubsection{Reified terms} *)
-
-Open Scope romega_scope.
-
-Fixpoint eq_term (t1 t2 : term) {struct t2} : bool :=
-  match t1, t2 with
-  | Tint st1, Tint st2 => beq st1 st2
-  | (st11 + st12), (st21 + st22) => eq_term st11 st21 && eq_term st12 st22
-  | (st11 * st12), (st21 * st22) => eq_term st11 st21 && eq_term st12 st22
-  | (st11 - st12), (st21 - st22) => eq_term st11 st21 && eq_term st12 st22
-  | (- st1), (- st2) => eq_term st1 st2
-  | [st1], [st2] => beq_nat st1 st2
-  | _, _ => false
-  end.
-
-Close Scope romega_scope.  
-
-Theorem eq_term_true : forall t1 t2 : term, eq_term t1 t2 = true -> t1 = t2.
-Proof.
- simple induction t1; intros until t2; case t2; simpl in *;
- try (intros; discriminate; fail); 
- [ intros; elim beq_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 beq_nat_true with (1 := H); trivial ].
-Qed.
-
-Ltac trivial_case := unfold not in |- *; intros; discriminate.
-
-Theorem eq_term_false :
- forall t1 t2 : term, eq_term t1 t2 = false -> t1 <> t2.
-Proof.
- simple induction t1;
- [ intros z t2; case t2; try trivial_case; simpl in |- *; unfold not in |- *;
-    intros; elim beq_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 beq_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. *)
-
-(* 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_eq_ind; intro Aux;
-   [ generalize (eq_term_true t1 t2 Aux); clear Aux
-   | generalize (eq_term_false t1 t2 Aux); clear Aux ].
-
-Ltac elim_beq t1 t2 :=
-  pattern (beq t1 t2) in |- *; apply bool_eq_ind; intro Aux;
-   [ generalize (beq_true t1 t2 Aux); clear Aux
-   | generalize (beq_false t1 t2 Aux); clear Aux ].
-
-Ltac elim_bgt t1 t2 := 
-  pattern (bgt t1 t2) in |- *; apply bool_eq_ind; intro Aux; 
-  [ generalize (bgt_true t1 t2 Aux); clear Aux
-  | generalize (bgt_false t1 t2 Aux); clear Aux ].
-
-
-(* \subsection{Interprétations}
-   \subsubsection{Interprétation des termes dans Z} *)
-
-Fixpoint interp_term (env : list int) (t : term) {struct t} : int :=
-  match t with
-  | Tint x => x
-  | (t1 + t2)%term => interp_term env t1 + interp_term env t2
-  | (t1 * t2)%term => interp_term env t1 * interp_term env t2
-  | (t1 - t2)%term => interp_term env t1 - interp_term env t2
-  | (- t)%term => - interp_term env t
-  | [n]%term => nth n env 0
-  end.
-
-(* \subsubsection{Interprétation des prédicats} *) 
-
-Fixpoint interp_proposition (envp : list Prop) (env : list int)
- (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
-  | TrueTerm => True
-  | FalseTerm => False
-  | Tnot p' => ~ interp_proposition envp env p'
-  | GeqTerm t1 t2 => interp_term env t1 >= interp_term env t2
-  | GtTerm t1 t2 => interp_term env t1 > interp_term env t2
-  | LtTerm t1 t2 => interp_term env t1 < interp_term env t2
-  | NeqTerm t1 t2 => (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 => nth 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 : list Prop) (env : list int) 
- (l : hyps) {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 (c : proposition) (envp : list Prop) 
- (env : list int) (l : hyps) {struct l} : Prop :=
-  match l with
-  | nil => interp_proposition envp env c
-  | p' :: l' =>
-      interp_proposition envp env p' -> interp_goal_concl c envp env l'
-  end.
-
-Notation interp_goal := (interp_goal_concl FalseTerm).
-
-(* Les théorèmes qui suivent assurent la correspondance entre les deux
-   interprétations. *)
-
-Theorem goal_to_hyps :
- forall (envp : list Prop) (env : list int) (l : hyps),
- (interp_hyps envp env l -> False) -> interp_goal envp env l.
-Proof.
- 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 : list Prop) (env : list int) (l : hyps),
- interp_goal envp env l -> interp_hyps envp env l -> False.
-Proof.
- 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 int) (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 : list Prop) (e : list int) (p1 : proposition),
-  interp_proposition ep e p1 -> interp_proposition ep e (f p1).
-
-Definition valid2 (f : proposition -> proposition -> proposition) :=
-  forall (ep : list Prop) (e : list int) (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 : hyps -> hyps) :=
-  forall (ep : list Prop) (e : list int) (lp : hyps),
-  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 : list Prop) (env : list int) (l : hyps) (a : hyps -> hyps),
-  valid_hyps a -> interp_goal ep env (a l) -> interp_goal ep env l.
-Proof.
- 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 : list Prop) (env : list int) 
- (l : lhyps) {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 : list Prop) (env : list int) 
- (l : lhyps) {struct l} : Prop :=
-  match l with
-  | nil => True
-  | h :: l' => interp_goal envp env h /\ interp_list_goal envp env l'
-  end.
-
-Theorem list_goal_to_hyps :
- forall (envp : list Prop) (env : list int) (l : lhyps),
- (interp_list_hyps envp env l -> False) -> interp_list_goal envp env l.
-Proof.
- 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 : list Prop) (env : list int) (l : lhyps),
- interp_list_goal envp env l -> interp_list_hyps envp env l -> False.
-Proof.
- 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 : hyps -> lhyps) :=
-  forall (ep : list Prop) (e : list int) (lp : hyps),
-  interp_hyps ep e lp -> interp_list_hyps ep e (f lp).
-
-Definition valid_list_goal (f : hyps -> lhyps) :=
-  forall (ep : list Prop) (e : list int) (lp : hyps),
-  interp_list_goal ep e (f lp) -> interp_goal ep e lp.
-
-Theorem goal_valid :
- forall f : hyps -> lhyps, valid_list_hyps f -> valid_list_goal f.
-Proof.
- 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 : list Prop) (e : list int) (l1 l2 : lhyps),
- interp_list_hyps ep e l1 \/ interp_list_hyps ep e l2 ->
- interp_list_hyps ep e (l1 ++ l2).
-Proof.
- 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 : hyps) := nth n l TrueTerm.
-
-Theorem nth_valid :
- forall (ep : list Prop) (e : list int) (i : nat) (l : hyps),
- interp_hyps ep e l -> interp_proposition ep e (nth_hyps i l).
-Proof.
- 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 : hyps) :=
-  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).
-Proof.
- 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 : hyps) {struct i} : hyps :=
-  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).
-Proof.
- 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
-  | (x + y)%term => (f x + y)%term
-  | (x * y)%term => (f x * y)%term
-  | (- x)%term => (- f x)%term
-  | x => x
-  end.
-
-Definition apply_right (f : term -> term) (t : term) :=
-  match t with
-  | (x + y)%term => (x + f y)%term
-  | (x * y)%term => (x * f y)%term
-  | x => x
-  end.
-
-Definition apply_both (f g : term -> term) (t : term) :=
-  match t with
-  | (x + y)%term => (f x + g y)%term
-  | (x * y)%term => (f x * g y)%term
-  | 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).
-Proof.
- 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).
-Proof.
- 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).
-Proof.
- 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)).
-Proof.
- 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 t with
-  (* Global *)
-  | (?X1 = ?X2) => loop X1 || loop X2
-  | (_ -> ?X1) => loop X1
-  (* Interpretations *)
-  | (interp_hyps _ _ ?X1) => loop X1
-  | (interp_list_hyps _ _ ?X1) => loop X1
-  | (interp_proposition _ _ ?X1) => loop X1
-  | (interp_term _ ?X1) => loop X1
-  (* Propositions *)
-  | (EqTerm ?X1 ?X2) => loop X1 || loop X2
-  | (LeqTerm ?X1 ?X2) => loop X1 || loop X2
-  (* Termes *)
-  | (?X1 + ?X2)%term => loop X1 || loop X2
-  | (?X1 - ?X2)%term => loop X1 || loop X2
-  | (?X1 * ?X2)%term => loop X1 || loop X2
-  | (- ?X1)%term => loop X1
-  | (Tint ?X1) => loop X1
-  (* Eliminations *)
-  | 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 => destruct X1; auto; Simplify
-  | match ?X1 with
-    | Tint x => _
-    | (x + x0)%term => _
-    | (x * x0)%term => _
-    | (x - x0)%term => _
-    | (- x)%term => _
-    | [x]%term => _
-    end => destruct X1; auto; Simplify
-  | (if beq ?X1 ?X2 then _ else _) =>
-      let H := fresh "H" in 
-      elim_beq X1 X2; intro H; try (rewrite H in *; clear H);
-       simpl in |- *; auto; Simplify
-  | (if bgt ?X1 ?X2 then _ else _) =>
-      let H := fresh "H" in 
-      elim_bgt X1 X2; intro H; simpl in |- *; auto; Simplify
-  | (if eq_term ?X1 ?X2 then _ else _) =>
-      let H := fresh "H" in 
-      elim_eq_term X1 X2; intro H; try (rewrite H in *; clear H); 
-       simpl in |- *; auto; Simplify
-  | (if _ && _ then _ else _) => rewrite andb_if; Simplify
-  | (if negb _ then _ else _) => rewrite negb_if; 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
-  | (n + (m + p))%term => (n + m + p)%term
-  | _ => t
-  end.
-
-Theorem Tplus_assoc_l_stable : term_stable Tplus_assoc_l.
-Proof.
- prove_stable Tplus_assoc_l (ring.(Radd_assoc)).
-Qed.
-
-Definition Tplus_assoc_r (t : term) :=
-  match t with
-  | (n + m + p)%term => (n + (m + p))%term
-  | _ => t
-  end.
-
-Theorem Tplus_assoc_r_stable : term_stable Tplus_assoc_r.
-Proof.
- prove_stable Tplus_assoc_r plus_assoc_reverse.
-Qed.
-
-Definition Tmult_assoc_r (t : term) :=
-  match t with
-  | (n * m * p)%term => (n * (m * p))%term
-  | _ => t
-  end.
-
-Theorem Tmult_assoc_r_stable : term_stable Tmult_assoc_r.
-Proof.
- prove_stable Tmult_assoc_r mult_assoc_reverse.
-Qed.
-
-Definition Tplus_permute (t : term) :=
-  match t with
-  | (n + (m + p))%term => (m + (n + p))%term
-  | _ => t
-  end.
-
-Theorem Tplus_permute_stable : term_stable Tplus_permute.
-Proof.
- prove_stable Tplus_permute plus_permute.
-Qed.
-
-Definition Tplus_comm (t : term) :=
-  match t with
-  | (x + y)%term => (y + x)%term
-  | _ => t
-  end.
-
-Theorem Tplus_comm_stable : term_stable Tplus_comm.
-Proof.
- prove_stable Tplus_comm plus_comm.
-Qed.
-
-Definition Tmult_comm (t : term) :=
-  match t with
-  | (x * y)%term => (y * x)%term
-  | _ => t
-  end.
-
-Theorem Tmult_comm_stable : term_stable Tmult_comm.
-Proof.
- prove_stable Tmult_comm mult_comm.
-Qed.
-
-Definition T_OMEGA10 (t : term) :=
-  match t with
-  | ((v * Tint c1 + l1) * Tint k1 + (v' * Tint c2 + l2) * Tint k2)%term =>
-      if eq_term v v' 
-      then (v * Tint (c1 * k1 + c2 * k2)%I + (l1 * Tint k1 + l2 * Tint k2))%term
-      else t
-  | _ => t
-  end.
-
-Theorem T_OMEGA10_stable : term_stable T_OMEGA10.
-Proof.
- prove_stable T_OMEGA10 OMEGA10.
-Qed.
-
-Definition T_OMEGA11 (t : term) :=
-  match t with
-  | ((v1 * Tint c1 + l1) * Tint k1 + l2)%term =>
-      (v1 * Tint (c1 * k1) + (l1 * Tint k1 + l2))%term
-  | _ => t
-  end.
-
-Theorem T_OMEGA11_stable : term_stable T_OMEGA11.
-Proof.
- prove_stable T_OMEGA11 OMEGA11.
-Qed.
-
-Definition T_OMEGA12 (t : term) :=
-  match t with
-  | (l1 + (v2 * Tint c2 + l2) * Tint k2)%term =>
-      (v2 * Tint (c2 * k2) + (l1 + l2 * Tint k2))%term
-  | _ => t
-  end.
-
-Theorem T_OMEGA12_stable : term_stable T_OMEGA12.
-Proof.
- prove_stable T_OMEGA12 OMEGA12.
-Qed. 
-
-Definition T_OMEGA13 (t : term) :=
-  match t with
-  | (v * Tint x + l1 + (v' * Tint x' + l2))%term =>
-      if eq_term v v' && beq x (-x') 
-      then (l1+l2)%term
-      else t
-  | _ => t
-  end.
-
-Theorem T_OMEGA13_stable : term_stable T_OMEGA13.
-Proof.
- unfold term_stable, T_OMEGA13 in |- *; intros; Simplify; simpl in |- *;
- apply OMEGA13.
-Qed.
-
-Definition T_OMEGA15 (t : term) :=
-  match t with
-  | (v * Tint c1 + l1 + (v' * Tint c2 + l2) * Tint k2)%term =>
-      if eq_term v v' 
-      then (v * Tint (c1 + c2 * k2)%I + (l1 + l2 * Tint k2))%term
-      else t
-  | _ => t
-  end.
-
-Theorem T_OMEGA15_stable : term_stable T_OMEGA15.
-Proof.
- prove_stable T_OMEGA15 OMEGA15.
-Qed.
-
-Definition T_OMEGA16 (t : term) :=
-  match t with
-  | ((v * Tint c + l) * Tint k)%term => (v * Tint (c * k) + l * Tint k)%term
-  | _ => t
-  end.
-
-
-Theorem T_OMEGA16_stable : term_stable T_OMEGA16.
-Proof.
- prove_stable T_OMEGA16 OMEGA16.
-Qed.
-
-Definition Tred_factor5 (t : term) :=
-  match t with
-  | (x * Tint c + y)%term => if beq c 0 then y else t
-  | _ => t
-  end.
-
-Theorem Tred_factor5_stable : term_stable Tred_factor5.
-Proof.
- prove_stable Tred_factor5 red_factor5.
-Qed.
-
-Definition Topp_plus (t : term) :=
-  match t with
-  | (- (x + y))%term => (- x + - y)%term
-  | _ => t
-  end.
-
-Theorem Topp_plus_stable : term_stable Topp_plus.
-Proof.
- prove_stable Topp_plus opp_plus_distr.
-Qed.
-
-
-Definition Topp_opp (t : term) :=
-  match t with
-  | (- - x)%term => x
-  | _ => t
-  end.
-
-Theorem Topp_opp_stable : term_stable Topp_opp.
-Proof.
- prove_stable Topp_opp opp_involutive.
-Qed.
-
-Definition Topp_mult_r (t : term) :=
-  match t with
-  | (- (x * Tint k))%term => (x * Tint (- k))%term
-  | _ => t
-  end.
-
-Theorem Topp_mult_r_stable : term_stable Topp_mult_r.
-Proof.
- prove_stable Topp_mult_r opp_mult_distr_r.
-Qed.
-
-Definition Topp_one (t : term) :=
-  match t with
-  | (- x)%term => (x * Tint (-(1)))%term
-  | _ => t
-  end.
-
-Theorem Topp_one_stable : term_stable Topp_one.
-Proof.
- prove_stable Topp_one opp_eq_mult_neg_1.
-Qed.
-
-Definition Tmult_plus_distr (t : term) :=
-  match t with
-  | ((n + m) * p)%term => (n * p + m * p)%term
-  | _ => t
-  end.
-
-Theorem Tmult_plus_distr_stable : term_stable Tmult_plus_distr.
-Proof.
- prove_stable Tmult_plus_distr mult_plus_distr_r.
-Qed.
-
-Definition Tmult_opp_left (t : term) :=
-  match t with
-  | (- x * Tint y)%term => (x * Tint (- y))%term
-  | _ => t
-  end.
-
-Theorem Tmult_opp_left_stable : term_stable Tmult_opp_left.
-Proof.
- prove_stable Tmult_opp_left mult_opp_comm.
-Qed.
-
-Definition Tmult_assoc_reduced (t : term) :=
-  match t with
-  | (n * Tint m * Tint p)%term => (n * Tint (m * p))%term
-  | _ => t
-  end.
-
-Theorem Tmult_assoc_reduced_stable : term_stable Tmult_assoc_reduced.
-Proof.
- prove_stable Tmult_assoc_reduced mult_assoc_reverse.
-Qed.
-
-Definition Tred_factor0 (t : term) := (t * Tint 1)%term.
-
-Theorem Tred_factor0_stable : term_stable Tred_factor0.
-Proof.
- prove_stable Tred_factor0 red_factor0.
-Qed.
-
-Definition Tred_factor1 (t : term) :=
-  match t with
-  | (x + y)%term =>
-      if eq_term x y 
-      then (x * Tint 2)%term
-      else t 
-  | _ => t
-  end.
-
-Theorem Tred_factor1_stable : term_stable Tred_factor1.
-Proof.
- prove_stable Tred_factor1 red_factor1.
-Qed.
-
-Definition Tred_factor2 (t : term) :=
-  match t with
-  | (x + y * Tint k)%term =>
-      if eq_term x y 
-      then (x * Tint (1 + k))%term
-      else t
-  | _ => t
-  end.
-
-Theorem Tred_factor2_stable : term_stable Tred_factor2.
-Proof.
- prove_stable Tred_factor2 red_factor2.
-Qed.
-
-Definition Tred_factor3 (t : term) :=
-  match t with
-  | (x * Tint k + y)%term =>
-      if eq_term x y 
-      then (x * Tint (1 + k))%term
-      else t
-  | _ => t
-  end.
-
-Theorem Tred_factor3_stable : term_stable Tred_factor3.
-Proof.
- prove_stable Tred_factor3 red_factor3.
-Qed.
-
-
-Definition Tred_factor4 (t : term) :=
-  match t with
-  | (x * Tint k1 + y * Tint k2)%term =>
-      if eq_term x y 
-      then (x * Tint (k1 + k2))%term
-      else t
-  | _ => t
-  end.
-
-Theorem Tred_factor4_stable : term_stable Tred_factor4.
-Proof.
- prove_stable Tred_factor4 red_factor4.
-Qed.
-
-Definition Tred_factor6 (t : term) := (t + Tint 0)%term.
-
-Theorem Tred_factor6_stable : term_stable Tred_factor6.
-Proof.
- prove_stable Tred_factor6 red_factor6.
-Qed.
-
-Definition Tminus_def (t : term) :=
-  match t with
-  | (x - y)%term => (x + - y)%term
-  | _ => t
-  end.
-
-Theorem Tminus_def_stable : term_stable Tminus_def.
-Proof.
- prove_stable Tminus_def minus_def.
-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
-  | (x + y)%term =>
-      match reduce x with
-      | Tint x' =>
-          match reduce y with
-          | Tint y' => Tint (x' + y')
-          | y' => (Tint x' + y')%term
-          end
-      | x' => (x' + reduce y)%term
-      end
-  | (x * y)%term =>
-      match reduce x with
-      | Tint x' =>
-          match reduce y with
-          | Tint y' => Tint (x' * y')
-          | y' => (Tint x' * y')%term
-          end
-      | x' => (x' * reduce y)%term
-      end
-  | (x - y)%term =>
-      match reduce x with
-      | Tint x' =>
-          match reduce y with
-          | Tint y' => Tint (x' - y')
-          | y' => (Tint x' - y')%term
-          end
-      | x' => (x' - reduce y)%term
-      end
-  | (- x)%term =>
-      match reduce x with
-      | Tint x' => Tint (- x')
-      | x' => (- x')%term
-      end
-  | _ => t
-  end.
-
-Theorem reduce_stable : term_stable reduce.
-Proof.
- 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).
-Proof.
- 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 annihilation} *)
-(* 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).
-Proof.
- 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 affines 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).
-Proof.
- 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).
-Proof.
- 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).
-Proof.
- 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_COMM => Tplus_comm
-  | 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_COMM => Tmult_comm
-  end.
-
-Theorem rewrite_stable : forall s : step, term_stable (rewrite s).
-Proof.
- 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_comm_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_comm_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 : hyps) :=
-  match nth_hyps i h with
-  | EqTerm (Tint Nul) (Tint n) =>
-      if beq n Nul then h else absurd
-  | _ => h
-  end.
-
-Theorem constant_not_nul_valid :
- forall i : nat, valid_hyps (constant_not_nul i).
-Proof.
- unfold valid_hyps, constant_not_nul in |- *; intros;
- generalize (nth_valid ep e i lp); Simplify; simpl in |- *. 
- 
- elim_beq i1 i0; auto; simpl in |- *; intros H1 H2; 
- elim H1; symmetry  in |- *; auto.
-Qed.   
-
-(* \paragraph{[O_CONSTANT_NEG]} *)
-
-Definition constant_neg (i : nat) (h : hyps) :=
-  match nth_hyps i h with
-  | LeqTerm (Tint Nul) (Tint Neg) => 
-     if bgt Nul Neg then absurd else h
-  | _ => h
-  end.
-
-Theorem constant_neg_valid : forall i : nat, valid_hyps (constant_neg i).
-Proof.
- unfold valid_hyps, constant_neg in |- *; intros;
- generalize (nth_valid ep e i lp); Simplify; simpl in |- *.
- rewrite gt_lt_iff in H0; rewrite le_lt_iff; intuition.
-Qed.
-
-(* \paragraph{[NOT_EXACT_DIVIDE]} *)
-Definition not_exact_divide (k1 k2 : int) (body : term) 
-  (t i : nat) (l : hyps) :=
-  match nth_hyps i l with
-  | EqTerm (Tint Nul) b =>
-      if beq Nul 0 && 
-         eq_term (scalar_norm_add t (body * Tint k1 + Tint k2)%term) b && 
-         bgt k2 0 && 
-         bgt k1 k2 
-      then absurd
-      else l
-  | _ => l
-  end.
-
-Theorem not_exact_divide_valid :
- forall (k1 k2 : int) (body : term) (t i : nat),
- valid_hyps (not_exact_divide k1 k2 body t i).
-Proof.
- unfold valid_hyps, not_exact_divide in |- *; intros;
- generalize (nth_valid ep e i lp); Simplify.
- rewrite (scalar_norm_add_stable t e), <-H1.
- do 2 rewrite <- scalar_norm_add_stable; simpl in *; intros.
- absurd (interp_term e body * k1 + k2 = 0); 
- [ now apply OMEGA4 | symmetry; auto ].
-Qed.
-
-(* \paragraph{[O_CONTRADICTION]} *)
-
-Definition contradiction (t i j : nat) (l : hyps) :=
-  match nth_hyps i l with
-  | LeqTerm (Tint Nul) b1 =>
-      match nth_hyps j l with
-      | LeqTerm (Tint Nul') b2 =>
-          match fusion_cancel t (b1 + b2)%term with
-          | Tint k => if beq Nul 0 && beq Nul' 0 && bgt 0 k 
-                      then absurd 
-                      else l
-          | _ => l
-          end
-      | _ => l
-      end
-  | _ => l
-  end.
-
-Theorem contradiction_valid :
- forall t i j : nat, valid_hyps (contradiction t i j).
-Proof.
- 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; case (nth_hyps j l); 
- auto; intros t3 t4; case t3; auto; 
- simpl in |- *; intros z z' H1 H2;
- generalize (refl_equal (interp_term e (fusion_cancel t (t2 + t4)%term)));
- pattern (fusion_cancel t (t2 + t4)%term) at 2 3 in |- *;
- case (fusion_cancel t (t2 + t4)%term); simpl in |- *; 
- auto; intro k; elim (fusion_cancel_stable t); simpl in |- *.
- Simplify; intro H3.
- generalize (OMEGA2 _ _ H2 H1); rewrite H3. 
- rewrite gt_lt_iff in H0; rewrite le_lt_iff; intuition.
-Qed.
-
-(* \paragraph{[O_NEGATE_CONTRADICT]} *)
-
-Definition negate_contradict (i1 i2 : nat) (h : hyps) :=
-  match nth_hyps i1 h with
-  | EqTerm (Tint Nul) b1 =>
-      match nth_hyps i2 h with
-      | NeqTerm (Tint Nul') b2 =>
-          if beq Nul 0 && beq Nul' 0 && eq_term b1 b2 
-          then absurd 
-          else h
-      | _ => h
-      end
-  | NeqTerm (Tint Nul) b1 =>
-      match nth_hyps i2 h with
-      | EqTerm (Tint Nul') b2 =>
-          if beq Nul 0 && beq Nul' 0 && eq_term b1 b2 
-          then absurd 
-          else h 
-      | _ => h
-      end
-  | _ => h
-  end.
-
-Definition negate_contradict_inv (t i1 i2 : nat) (h : hyps) :=
-  match nth_hyps i1 h with
-  | EqTerm (Tint Nul) b1 =>
-      match nth_hyps i2 h with
-      | NeqTerm (Tint Nul') b2 =>
-          if beq Nul 0 && beq Nul' 0 && 
-             eq_term b1 (scalar_norm t (b2 * Tint (-(1)))%term)
-          then absurd
-          else h
-      | _ => h
-      end
-  | NeqTerm (Tint Nul) b1 =>
-      match nth_hyps i2 h with
-      | EqTerm (Tint Nul') b2 =>
-          if beq Nul 0 && beq Nul' 0 && 
-             eq_term b1 (scalar_norm t (b2 * Tint (-(1)))%term)
-          then absurd
-          else h
-      | _ => h
-      end
-  | _ => h
-  end.
-
-Theorem negate_contradict_valid :
- forall i j : nat, valid_hyps (negate_contradict i j).
-Proof.
- 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; auto; case (nth_hyps j l); 
- auto; intros t3 t4; case t3; auto; intros z'; 
- auto; simpl in |- *; intros H1 H2; Simplify.
-Qed.
-
-Theorem negate_contradict_inv_valid :
- forall t i j : nat, valid_hyps (negate_contradict_inv t i j).
-Proof.
- 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; auto; case (nth_hyps j l); 
- auto; intros t3 t4; case t3; auto; intros z'; 
- auto; simpl in |- *; intros H1 H2; Simplify; 
- [
- rewrite <- scalar_norm_stable in H2; simpl in *;
- elim (mult_integral (interp_term e t4) (-(1))); intuition;
- elim minus_one_neq_zero; auto
- | 
- elim H2; clear H2;
- rewrite <- scalar_norm_stable; simpl in *;
- now rewrite <- H1, mult_0_l
- ].
-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 : int) (trace : list t_fusion)
-  (prop1 prop2 : proposition) :=
-  match prop1 with
-  | EqTerm (Tint Null) b1 =>
-      match prop2 with
-      | EqTerm (Tint Null') b2 =>
-          if beq Null 0 && beq Null' 0 
-          then EqTerm (Tint 0) (fusion trace (b1 * Tint k1 + b2 * Tint k2)%term)
-          else TrueTerm
-      | LeqTerm (Tint Null') b2 =>
-          if beq Null 0 && beq Null' 0 && bgt k2 0 
-          then LeqTerm (Tint 0)
-                (fusion trace (b1 * Tint k1 + b2 * Tint k2)%term)
-          else TrueTerm
-      | _ => TrueTerm
-      end
-  | LeqTerm (Tint Null) b1 =>
-      if beq Null 0 && bgt k1 0
-      then match prop2 with
-          | EqTerm (Tint Null') b2 =>
-              if beq Null' 0 then 
-              LeqTerm (Tint 0)
-                (fusion trace (b1 * Tint k1 + b2 * Tint k2)%term)
-              else TrueTerm  
-          | LeqTerm (Tint Null') b2 =>
-              if beq Null' 0 && bgt k2 0 
-              then LeqTerm (Tint 0)
-                    (fusion trace (b1 * Tint k1 + b2 * Tint k2)%term)
-              else TrueTerm
-          | _ => TrueTerm
-          end
-      else TrueTerm 
-  | NeqTerm (Tint Null) b1 =>
-      match prop2 with
-      | EqTerm (Tint Null') b2 =>
-          if beq Null 0 && beq Null' 0 && (negb (beq k1 0))
-          then NeqTerm (Tint 0)
-                 (fusion trace (b1 * Tint k1 + b2 * Tint k2)%term)
-          else TrueTerm
-      | _ => TrueTerm
-      end
-  | _ => TrueTerm
-  end.
-
-
-Theorem sum_valid :
- forall (k1 k2 : int) (t : list t_fusion), valid2 (sum k1 k2 t).
-Proof.
- 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 plus_comm; apply sum2; try assumption; apply sum4; assumption
- | apply sum3; try assumption; apply sum4; assumption
- | apply sum5; auto ].
-Qed.
-
-(* \paragraph{[O_EXACT_DIVIDE]}
-   c'est une oper1 valide mais on préfère une substitution a ce point la *)
-
-Definition exact_divide (k : int) (body : term) (t : nat)
-  (prop : proposition) :=
-  match prop with
-  | EqTerm (Tint Null) b =>
-      if beq Null 0 && 
-         eq_term (scalar_norm t (body * Tint k)%term) b && 
-         negb (beq k 0) 
-      then EqTerm (Tint 0) body
-      else TrueTerm
-  | NeqTerm (Tint Null) b =>
-      if beq Null 0 && 
-         eq_term (scalar_norm t (body * Tint k)%term) b &&
-         negb (beq k 0)
-      then NeqTerm (Tint 0) body
-      else TrueTerm
-  | _ => TrueTerm
-  end.
-
-Theorem exact_divide_valid :
- forall (k : int) (t : term) (n : nat), valid1 (exact_divide k t n).
-Proof.
- unfold valid1, exact_divide in |- *; intros k1 k2 t ep e p1; 
- Simplify; simpl; auto; subst; 
- rewrite <- scalar_norm_stable; simpl; intros;
- [ destruct (mult_integral _ _ (sym_eq H0)); intuition
- | contradict H0; rewrite <- H0, mult_0_l; auto
- ].
-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 : int) (body : term) 
-  (t : nat) (prop : proposition) :=
-  match prop with
-  | LeqTerm (Tint Null) b =>
-      if beq Null 0 && 
-         eq_term (scalar_norm_add t (body * Tint k1 + Tint k2)%term) b &&
-         bgt k1 0 && 
-         bgt k1 k2 
-      then LeqTerm (Tint 0) body 
-      else prop
-  | _ => prop
-  end.
-
-Theorem divide_and_approx_valid :
- forall (k1 k2 : int) (body : term) (t : nat),
- valid1 (divide_and_approx k1 k2 body t).
-Proof.
- unfold valid1, divide_and_approx in |- *; intros k1 k2 body t ep e p1;
- Simplify; simpl; auto; subst;
- elim (scalar_norm_add_stable t e); simpl in |- *.
- intro H2; apply mult_le_approx with (3 := H2); assumption.
-Qed.
-
-(* \paragraph{[MERGE_EQ]} *)
-
-Definition merge_eq (t : nat) (prop1 prop2 : proposition) :=
-  match prop1 with
-  | LeqTerm (Tint Null) b1 =>
-      match prop2 with
-      | LeqTerm (Tint Null') b2 =>
-          if beq Null 0 && beq Null' 0 &&
-             eq_term b1 (scalar_norm t (b2 * Tint (-(1)))%term) 
-          then EqTerm (Tint 0) b1
-          else TrueTerm
-      | _ => TrueTerm
-      end
-  | _ => TrueTerm
-  end.
-
-Theorem merge_eq_valid : forall n : nat, valid2 (merge_eq n).
-Proof.
- 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 opp_eq_mult_neg_1; trivial ].
-Qed.
-
-
-
-(* \paragraph{[O_CONSTANT_NUL]} *)
-
-Definition constant_nul (i : nat) (h : hyps) :=
-  match nth_hyps i h with
-  | NeqTerm (Tint Null) (Tint Null') => 
-      if beq Null Null' then absurd else h 
-  | _ => h
-  end.
-
-Theorem constant_nul_valid : forall i : nat, valid_hyps (constant_nul i).
-Proof.
- unfold valid_hyps, constant_nul in |- *; intros;
- generalize (nth_valid ep e i lp); Simplify; simpl in |- *;
- intro H1; absurd (0 = 0); intuition.
-Qed.
-
-(* \paragraph{[O_STATE]} *)
-
-Definition state (m : int) (s : step) (prop1 prop2 : proposition) :=
-  match prop1 with
-  | EqTerm (Tint Null) b1 =>
-      match prop2 with
-      | EqTerm b2 b3 =>
-          if beq Null 0 
-          then EqTerm (Tint 0) (rewrite s (b1 + (- b3 + b2) * Tint m)%term)
-          else TrueTerm
-      | _ => TrueTerm
-      end
-  | _ => TrueTerm
-  end.
-
-Theorem state_valid : forall (m : int) (s : step), valid2 (state m s).
-Proof.
- 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.
- now rewrite H2, plus_opp_l, plus_0_l, mult_0_l.
-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 : hyps -> lhyps) 
-  (l : hyps) :=
-  match nth_hyps i l with
-  | NeqTerm (Tint Null) b1 =>
-      if beq Null 0 then 
-      f1 (LeqTerm (Tint 0) (add_norm t (b1 + Tint (-(1)))%term) :: l) ++
-      f2
-        (LeqTerm (Tint 0)
-           (scalar_norm_add t (b1 * Tint (-(1)) + Tint (-(1)))%term) :: l)
-       else l :: nil
-  | _ => l :: nil
-  end.
-
-Theorem split_ineq_valid :
- forall (i t : nat) (f1 f2 : hyps -> lhyps),
- valid_list_hyps f1 ->
- valid_list_hyps f2 -> valid_list_hyps (split_ineq i t f1 f2).
-Proof.
- 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; simpl in |- *; auto; intro H3.
- Simplify.
- 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 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 : hyps) {struct t} : lhyps :=
-  match t with
-  | O_CONSTANT_NOT_NUL n => singleton (constant_not_nul n l)
-  | O_CONSTANT_NEG n => singleton (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 =>
-      singleton (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 => singleton (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 => singleton (constant_nul i l)
-  | O_NEGATE_CONTRADICT i j => singleton (negate_contradict i j l)
-  | O_NEGATE_CONTRADICT_INV t i j =>
-      singleton (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).
-Proof.
- 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 i i0 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 (t1 + - t2)%term)
-  | LeqTerm t1 t2 => LeqTerm (Tint 0) (rewrite s (t2 + - t1)%term)
-  | GeqTerm t1 t2 => LeqTerm (Tint 0) (rewrite s (t1 + - t2)%term)
-  | LtTerm t1 t2 => LeqTerm (Tint 0) (rewrite s (t2 + Tint (-(1)) + - t1)%term)
-  | GtTerm t1 t2 => LeqTerm (Tint 0) (rewrite s (t1 + Tint (-(1)) + - t2)%term)
-  | NeqTerm t1 t2 => NeqTerm (Tint 0) (rewrite s (t1 + - t2)%term)
-  | p => p
-  end.
-
-Theorem move_right_valid : forall s : step, valid1 (move_right s).
-Proof.
- unfold valid1, move_right in |- *; intros s ep e p; Simplify; simpl in |- *;
- elim (rewrite_stable s e); simpl in |- *;
- [ symmetry  in |- *; apply egal_left; assumption
- | intro; apply le_left; assumption
- | intro; apply le_left; rewrite <- ge_le_iff; assumption
- | intro; apply lt_left; rewrite <- gt_lt_iff; assumption
- | intro; apply lt_left; assumption
- | intro; apply ne_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).
-Proof.
- 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 : hyps) {struct l} : hyps :=
-  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).
-Proof.
- 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 : list Prop) (env : list int) (l : hyps),
- interp_goal ep env (do_normalize_list s 0 l) -> interp_goal ep env l.
-Proof.
- 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 : list Prop) (env : list int) (l : hyps),
- interp_list_goal ep env (execute_omega t l) -> interp_goal ep env l.
-Proof.
- intros; apply (goal_valid (execute_omega t) (omega_valid t) ep env l H).
-Qed.
-
-
-Theorem append_goal :
- forall (ep : list Prop) (e : list int) (l1 l2 : lhyps),
- interp_list_goal ep e l1 /\ interp_list_goal ep e l2 ->
- interp_list_goal ep e (l1 ++ l2).
-Proof.
- 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.
-
-(* 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 : list Prop) (e : list int) (p : proposition),
- decidability p = true -> decidable (interp_proposition ep e p).
-Proof.
- simple induction p; simpl in |- *; intros;
- [ apply dec_eq
- | apply dec_le
- | left; auto
- | right; unfold not in |- *; auto
- | apply dec_not; auto
- | apply dec_ge
- | apply dec_gt
- | apply dec_lt
- | apply dec_ne
- | 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 : list Prop) (env : list int) 
- (c : proposition) (l : hyps) {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 : list Prop) (e : list int) 
-  (lc : hyps * 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 : list Prop) (e : list int) (l : hyps) (c : proposition),
- (interp_hyps ep e l -> interp_proposition ep e c) -> interp_full ep e (l, c).
-Proof.
- 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 : hyps * 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 : list Prop) (e : list int) (lc : hyps * proposition),
- interp_goal ep e (to_contradict lc) -> interp_full ep e lc.
-Proof.
- intros ep e lc; case lc; intros l c; simpl in |- *;
- pattern (decidability c) in |- *; apply bool_eq_ind;
- [ 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 : hyps) {struct nn} : lhyps :=
-  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 : list Prop) (e : list int) (p : proposition) (l : lhyps),
- interp_proposition ep e p ->
- interp_list_hyps ep e l -> interp_list_hyps ep e (map_cons _ p l).
-Proof.
- 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).
-Proof.
- 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_eq_ind; 
-             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_eq_ind; 
-             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_eq_ind; 
-          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 : list Prop) (e : list int) (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).
-Proof.
- 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).
-Proof.
- 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).
-Proof.
- 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 |- *;
-      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 |- *;
-      generalize (dec_gt (interp_term e t1) (interp_term e t2));
-      unfold decidable in |- *; rewrite le_lt_iff, <- gt_lt_iff; tauto
-   | intros t1 t2; elim (H ep e (LtTerm t1 t2)); simpl in |- *;
-      generalize (dec_lt (interp_term e t1) (interp_term e t2));
-      unfold decidable in |- *; rewrite ge_le_iff, le_lt_iff; tauto
-   | intros t1 t2; elim (H ep e (LeqTerm t1 t2)); simpl in |- *;
-      generalize (dec_gt (interp_term e t1) (interp_term e t2));
-      unfold decidable in |- *; repeat rewrite le_lt_iff;
-      repeat rewrite gt_lt_iff; tauto
-   | intros t1 t2; elim (H ep e (GeqTerm t1 t2)); simpl in |- *;
-      generalize (dec_lt (interp_term e t1) (interp_term e t2));
-      unfold decidable in |- *; repeat rewrite ge_le_iff;
-      repeat rewrite le_lt_iff; 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; tauto ]).
-Qed.
-
-Theorem move_right_stable : forall s : step, prop_stable (move_right s).
-Proof.
- 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 egal_left; assumption
-    | intro; apply le_left; assumption
-    | intro; apply le_left; rewrite <- ge_le_iff; assumption
-    | intro; apply lt_left; rewrite <- gt_lt_iff; assumption
-    | intro; apply lt_left; assumption
-    | intro; apply ne_left_2; assumption ]
- | case p; simpl in |- *; intros; auto; generalize H; elim (rewrite_stable s);
-    simpl in |- *; intro H1;
-    [ rewrite (plus_0_r_reverse (interp_term e t0)); rewrite H1;
-       rewrite plus_permute; rewrite plus_opp_r; 
-       rewrite plus_0_r; trivial
-    | apply (fun a b => plus_le_reg_r a b (- interp_term e t));
-       rewrite plus_opp_r; assumption
-    | rewrite ge_le_iff;
-       apply (fun a b => plus_le_reg_r a b (- interp_term e t0));
-       rewrite plus_opp_r; assumption
-    | rewrite gt_lt_iff; apply lt_left_inv; assumption
-    | apply lt_left_inv; assumption
-    | unfold not in |- *; intro H2; apply H1;
-       rewrite H2; rewrite plus_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).
-Proof.
- 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 : hyps) {struct l} : hyps :=
-  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).
-Proof.
- 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 : list Prop) (env : list int) (l : hyps),
- interp_goal ep env (normalize_hyps s l) -> interp_goal ep env l.
-Proof.
- 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 : list Prop) (e : list int) (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).
-Proof.
- 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_eq_ind;
-       [ 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_eq_ind;
-          [ 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 : hyps) {struct s} : lhyps :=
-  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
-      | Timp x y => 
-          if decidability x then 
-            decompose_solve s1 (Tnot x :: h) ++ decompose_solve s2 (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).
-Proof.
- 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_eq_ind;
-          [ 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 ] 
-       | intros p1 p2 H2;
-         pattern (decidability p1) in |- *; apply bool_eq_ind;
-          [ 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 ] ]
-    | 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 : lhyps -> lhyps) :=
-  forall (ep : list Prop) (e : list int) (lp : lhyps),
-  interp_list_hyps ep e lp -> interp_list_hyps ep e (f lp).
-
-Fixpoint reduce_lhyps (lp : lhyps) : lhyps :=
-  match lp with
-  | (FalseTerm :: nil) :: lp' => reduce_lhyps lp'
-  | x :: lp' => x :: reduce_lhyps lp'
-  | nil => nil (A:=hyps)
-  end.
-
-Theorem reduce_lhyps_valid : valid_lhyps reduce_lhyps.
-Proof.
- 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 : list Prop) (env : list int) (l : lhyps),
- interp_list_goal envp env (reduce_lhyps l) -> interp_list_goal envp env l.
-Proof.
- 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 : list Prop) (env : list int) (c : proposition) (l : hyps),
-  interp_goal envp env (concl_to_hyp c :: l) ->
-  interp_goal_concl c envp env l.
-Proof.
- 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_eq_ind;
-    [ 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 : hyps) :=
-  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 : list Prop) 
-   (env : list int) (c : proposition) (l : hyps),
- interp_list_goal envp env (omega_tactic t1 t2 c l) ->
- interp_goal_concl c envp env l.
-Proof.
- 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.
-
-End IntOmega.
-
-(* For now, the above modular construction is instanciated on Z, 
-   in order to retrieve the initial ROmega. *)
-
-Module ZOmega := IntOmega(Z_as_Int).
diff --git a/contrib/romega/const_omega.ml b/contrib/romega/const_omega.ml
deleted file mode 100644
index bdec6bf4..00000000
--- a/contrib/romega/const_omega.ml
+++ /dev/null
@@ -1,350 +0,0 @@
-(*************************************************************************
-
-   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
-  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 logic_dir = ["Coq";"Logic";"Decidable"]
-
-let coq_modules =
-  Coqlib.init_modules @ [logic_dir] @ Coqlib.arith_modules @ Coqlib.zarith_base_modules
-    @ [["Coq"; "Lists"; "List"]]
-    @ [module_refl_path]
-    @ [module_refl_path@["ZOmega"]]
-
-let constant = Coqlib.gen_constant_in_modules "Omega" coq_modules
-
-(* 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_I = lazy(constant "I")
-
-(* ReflOmegaCore/ZOmega *)
-
-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_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_proposition = lazy (constant  "proposition")
-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")
-
-(* 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_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_comm = lazy (constant  "C_PLUS_COMM")
-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_comm = lazy (constant  "C_MULT_COMM")
-
-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_interp_sequent = lazy (constant  "interp_goal_concl")
-let coq_do_omega = lazy (constant  "do_omega")
-
-(* \subsection{Construction d'expressions} *)
-
-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)
-
-(* Nat *)
-
-let coq_S = lazy(constant "S")
-let coq_O = lazy(constant "O")
-
-let rec mk_nat = function
-  | 0 -> Lazy.force coq_O
-  | n -> Term.mkApp (Lazy.force coq_S, [| mk_nat (n-1) |])
-
-(* Lists *)
-
-let coq_cons =  lazy (constant "cons")
-let coq_nil =  lazy (constant "nil")
-
-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 = mk_list Term.mkProp l
-
-let mk_shuffle_list l = mk_list (Lazy.force coq_t_fusion) l
-
-
-type parse_term = 
-  | Tplus of Term.constr * Term.constr 
-  | Tmult of Term.constr * Term.constr
-  | Tminus of Term.constr * Term.constr
-  | Topp of Term.constr
-  | Tsucc of Term.constr
-  | Tnum of Bigint.bigint
-  | Tother 
-
-type parse_rel = 
-  | Req of Term.constr * Term.constr
-  | Rne of Term.constr * Term.constr
-  | Rlt of Term.constr * Term.constr
-  | Rle of Term.constr * Term.constr
-  | Rgt of Term.constr * Term.constr
-  | Rge of Term.constr * Term.constr
-  | Rtrue
-  | Rfalse
-  | Rnot of Term.constr
-  | Ror of Term.constr * Term.constr
-  | Rand of Term.constr * Term.constr
-  | Rimp of Term.constr * Term.constr
-  | Riff of Term.constr * Term.constr
-  | Rother
-
-let parse_logic_rel c = 
- try match destructurate c with
-   | Kapp("True",[]) -> Rtrue
-   | Kapp("False",[]) -> Rfalse
-   | Kapp("not",[t]) -> Rnot t
-   | Kapp("or",[t1;t2]) -> Ror (t1,t2) 
-   | Kapp("and",[t1;t2]) -> Rand (t1,t2)
-   | Kimp(t1,t2) -> Rimp (t1,t2)
-   | Kapp("iff",[t1;t2]) -> Riff (t1,t2)
-   | _ -> Rother
- with e when Logic.catchable_exception e -> Rother
-
-
-module type Int = sig
-  val typ : Term.constr Lazy.t
-  val plus : Term.constr Lazy.t 
-  val mult : Term.constr Lazy.t
-  val opp : Term.constr Lazy.t
-  val minus : Term.constr Lazy.t
-
-  val mk : Bigint.bigint -> Term.constr
-  val parse_term : Term.constr -> parse_term
-  val parse_rel : Proof_type.goal Tacmach.sigma -> Term.constr -> parse_rel
-  (* check whether t is built only with numbers and + * - *)
-  val is_scalar : Term.constr -> bool 
-end
-
-module Z : Int = struct 
-
-let typ = lazy (constant "Z")
-let plus = lazy (constant "Zplus")
-let mult = lazy (constant  "Zmult")
-let opp = lazy (constant "Zopp")
-let minus = lazy (constant "Zminus")
-
-let coq_xH = lazy (constant "xH")
-let coq_xO = lazy (constant "xO")
-let coq_xI = lazy (constant "xI")
-let coq_Z0 = lazy (constant "Z0")
-let coq_Zpos = lazy (constant "Zpos")
-let coq_Zneg = lazy (constant "Zneg")
-
-let recognize t =
-  let rec loop t =
-    let f,l = dest_const_apply t in
-    match Names.string_of_id f,l with
-       "xI",[t] -> Bigint.add Bigint.one (Bigint.mult Bigint.two (loop t))
-     | "xO",[t] -> Bigint.mult Bigint.two (loop t)
-     | "xH",[] -> Bigint.one
-     | _ -> 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] -> Bigint.neg (loop t)
-      | "Z0",[] -> Bigint.zero
-      | _ -> failwith "not a number";;
-
-let rec mk_positive n = 
-  if n=Bigint.one then Lazy.force coq_xH 
-  else
-    let (q,r) = Bigint.euclid n Bigint.two in
-    Term.mkApp
-      ((if r = Bigint.zero then Lazy.force coq_xO else Lazy.force coq_xI),
-       [| mk_positive q |]) 
-
-let mk_Z n =
-  if n = Bigint.zero then Lazy.force coq_Z0 
-  else if Bigint.is_strictly_pos n then
-    Term.mkApp (Lazy.force coq_Zpos, [| mk_positive n |])
-  else
-    Term.mkApp (Lazy.force coq_Zneg, [| mk_positive (Bigint.neg n) |])
-
-let mk = mk_Z
-
-let parse_term t = 
-  try match destructurate t with
-    | Kapp("Zplus",[t1;t2]) -> Tplus (t1,t2)
-    | Kapp("Zminus",[t1;t2]) -> Tminus (t1,t2)
-    | Kapp("Zmult",[t1;t2]) -> Tmult (t1,t2)
-    | Kapp("Zopp",[t]) -> Topp t
-    | Kapp("Zsucc",[t]) -> Tsucc t
-    | Kapp("Zpred",[t]) -> Tplus(t, mk_Z (Bigint.neg Bigint.one))
-    | Kapp(("Zpos"|"Zneg"|"Z0"),_) -> 
-	(try Tnum (recognize t) with _ -> Tother)
-    | _ -> Tother
-  with e when Logic.catchable_exception e -> Tother
-    
-let parse_rel gl t = 
-  try match destructurate t with 
-    | Kapp("eq",[typ;t1;t2]) 
-      when destructurate (Tacmach.pf_nf gl typ) = Kapp("Z",[]) -> Req (t1,t2)
-    | Kapp("Zne",[t1;t2]) -> Rne (t1,t2)
-    | Kapp("Zle",[t1;t2]) -> Rle (t1,t2)
-    | Kapp("Zlt",[t1;t2]) -> Rlt (t1,t2)
-    | Kapp("Zge",[t1;t2]) -> Rge (t1,t2)
-    | Kapp("Zgt",[t1;t2]) -> Rgt (t1,t2)
-    | _ -> parse_logic_rel t 
-  with e when Logic.catchable_exception e -> Rother
-
-let is_scalar t =
-  let rec aux t = match destructurate t with
-    | Kapp(("Zplus"|"Zminus"|"Zmult"),[t1;t2]) -> aux t1 & aux t2
-    | Kapp(("Zopp"|"Zsucc"|"Zpred"),[t]) -> aux t
-    | Kapp(("Zpos"|"Zneg"|"Z0"),_) -> let _ = recognize t in true
-    | _ -> false in
-  try aux t with _ -> false
-
-end
diff --git a/contrib/romega/const_omega.mli b/contrib/romega/const_omega.mli
deleted file mode 100644
index 0f00e918..00000000
--- a/contrib/romega/const_omega.mli
+++ /dev/null
@@ -1,176 +0,0 @@
-(*************************************************************************
-
-   PROJET RNRT Calife - 2001
-   Author: Pierre Crégut - France Télécom R&D
-   Licence : LGPL version 2.1
-
- *************************************************************************)
-
-
-(** Coq objects used in romega *)
-
-(* from Logic *)
-val coq_refl_equal : Term.constr lazy_t
-val coq_and : Term.constr lazy_t
-val coq_not : Term.constr lazy_t
-val coq_or : Term.constr lazy_t
-val coq_True : Term.constr lazy_t
-val coq_False : Term.constr lazy_t
-val coq_I : Term.constr lazy_t
-
-(* from ReflOmegaCore/ZOmega *)
-val coq_h_step : Term.constr lazy_t
-val coq_pair_step : Term.constr lazy_t
-val coq_p_left : Term.constr lazy_t
-val coq_p_right : Term.constr lazy_t
-val coq_p_invert : Term.constr lazy_t
-val coq_p_step : Term.constr lazy_t
-
-val coq_t_int : Term.constr lazy_t
-val coq_t_plus : Term.constr lazy_t
-val coq_t_mult : Term.constr lazy_t
-val coq_t_opp : Term.constr lazy_t
-val coq_t_minus : Term.constr lazy_t
-val coq_t_var : Term.constr lazy_t
-
-val coq_proposition : Term.constr lazy_t
-val coq_p_eq : Term.constr lazy_t
-val coq_p_leq : Term.constr lazy_t
-val coq_p_geq : Term.constr lazy_t
-val coq_p_lt : Term.constr lazy_t
-val coq_p_gt : Term.constr lazy_t
-val coq_p_neq : Term.constr lazy_t
-val coq_p_true : Term.constr lazy_t
-val coq_p_false : Term.constr lazy_t
-val coq_p_not : Term.constr lazy_t
-val coq_p_or : Term.constr lazy_t
-val coq_p_and : Term.constr lazy_t
-val coq_p_imp : Term.constr lazy_t
-val coq_p_prop : Term.constr lazy_t
-
-val coq_f_equal : Term.constr lazy_t
-val coq_f_cancel : Term.constr lazy_t
-val coq_f_left : Term.constr lazy_t
-val coq_f_right : Term.constr lazy_t
-
-val coq_c_do_both : Term.constr lazy_t
-val coq_c_do_left : Term.constr lazy_t
-val coq_c_do_right : Term.constr lazy_t
-val coq_c_do_seq : Term.constr lazy_t
-val coq_c_nop : Term.constr lazy_t
-val coq_c_opp_plus : Term.constr lazy_t
-val coq_c_opp_opp : Term.constr lazy_t
-val coq_c_opp_mult_r : Term.constr lazy_t
-val coq_c_opp_one : Term.constr lazy_t
-val coq_c_reduce : Term.constr lazy_t
-val coq_c_mult_plus_distr : Term.constr lazy_t
-val coq_c_opp_left : Term.constr lazy_t
-val coq_c_mult_assoc_r : Term.constr lazy_t
-val coq_c_plus_assoc_r : Term.constr lazy_t
-val coq_c_plus_assoc_l : Term.constr lazy_t
-val coq_c_plus_permute : Term.constr lazy_t
-val coq_c_plus_comm : Term.constr lazy_t
-val coq_c_red0 : Term.constr lazy_t
-val coq_c_red1 : Term.constr lazy_t
-val coq_c_red2 : Term.constr lazy_t
-val coq_c_red3 : Term.constr lazy_t
-val coq_c_red4 : Term.constr lazy_t
-val coq_c_red5 : Term.constr lazy_t
-val coq_c_red6 : Term.constr lazy_t
-val coq_c_mult_opp_left : Term.constr lazy_t
-val coq_c_mult_assoc_reduced : Term.constr lazy_t
-val coq_c_minus : Term.constr lazy_t
-val coq_c_mult_comm : Term.constr lazy_t
-
-val coq_s_constant_not_nul : Term.constr lazy_t
-val coq_s_constant_neg : Term.constr lazy_t
-val coq_s_div_approx : Term.constr lazy_t
-val coq_s_not_exact_divide : Term.constr lazy_t
-val coq_s_exact_divide : Term.constr lazy_t
-val coq_s_sum : Term.constr lazy_t
-val coq_s_state : Term.constr lazy_t
-val coq_s_contradiction : Term.constr lazy_t
-val coq_s_merge_eq : Term.constr lazy_t
-val coq_s_split_ineq : Term.constr lazy_t
-val coq_s_constant_nul : Term.constr lazy_t
-val coq_s_negate_contradict : Term.constr lazy_t
-val coq_s_negate_contradict_inv : Term.constr lazy_t
-
-val coq_direction : Term.constr lazy_t
-val coq_d_left : Term.constr lazy_t
-val coq_d_right : Term.constr lazy_t
-val coq_d_mono : Term.constr lazy_t
-
-val coq_e_split : Term.constr lazy_t
-val coq_e_extract : Term.constr lazy_t
-val coq_e_solve : Term.constr lazy_t
-
-val coq_interp_sequent : Term.constr lazy_t
-val coq_do_omega : Term.constr lazy_t
-
-(** Building expressions *)
-
-val do_left : Term.constr -> Term.constr
-val do_right : Term.constr -> Term.constr
-val do_both : Term.constr -> Term.constr -> Term.constr
-val do_seq : Term.constr -> Term.constr -> Term.constr
-val do_list : Term.constr list -> Term.constr
-
-val mk_nat : int -> Term.constr
-val mk_list : Term.constr -> Term.constr list -> Term.constr
-val mk_plist : Term.types list -> Term.types
-val mk_shuffle_list : Term.constr list -> Term.constr
-
-(** Analyzing a coq term *)
-
-(* The generic result shape of the analysis of a term.
-   One-level depth, except when a number is found *)
-type parse_term =
-    Tplus of Term.constr * Term.constr
-  | Tmult of Term.constr * Term.constr
-  | Tminus of Term.constr * Term.constr
-  | Topp of Term.constr
-  | Tsucc of Term.constr
-  | Tnum of Bigint.bigint
-  | Tother
-
-(* The generic result shape of the analysis of a relation.
-   One-level depth. *)
-type parse_rel =
-    Req of Term.constr * Term.constr
-  | Rne of Term.constr * Term.constr
-  | Rlt of Term.constr * Term.constr
-  | Rle of Term.constr * Term.constr
-  | Rgt of Term.constr * Term.constr
-  | Rge of Term.constr * Term.constr
-  | Rtrue
-  | Rfalse
-  | Rnot of Term.constr
-  | Ror of Term.constr * Term.constr
-  | Rand of Term.constr * Term.constr
-  | Rimp of Term.constr * Term.constr
-  | Riff of Term.constr * Term.constr
-  | Rother
-
-(* A module factorizing what we should now about the number representation *)
-module type Int =
-  sig
-    (* the coq type of the numbers *)
-    val typ : Term.constr Lazy.t
-    (* the operations on the numbers *)
-    val plus : Term.constr Lazy.t
-    val mult : Term.constr Lazy.t
-    val opp : Term.constr Lazy.t
-    val minus : Term.constr Lazy.t
-    (* building a coq number *)
-    val mk : Bigint.bigint -> Term.constr
-    (* parsing a term (one level, except if a number is found) *)
-    val parse_term : Term.constr -> parse_term
-    (* parsing a relation expression, including = < <= >= > *)
-    val parse_rel : Proof_type.goal Tacmach.sigma -> Term.constr -> parse_rel
-    (* Is a particular term only made of numbers and + * - ? *) 
-    val is_scalar : Term.constr -> bool
-  end
-
-(* Currently, we only use Z numbers *)
-module Z : Int
diff --git a/contrib/romega/g_romega.ml4 b/contrib/romega/g_romega.ml4
deleted file mode 100644
index 39b6c210..00000000
--- a/contrib/romega/g_romega.ml4
+++ /dev/null
@@ -1,42 +0,0 @@
-(*************************************************************************
-
-   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
-open Refiner
-
-let romega_tactic l = 
-  let tacs = List.map 
-    (function  
-       | "nat" -> Tacinterp.interp <:tactic<zify_nat>>
-       | "positive" -> Tacinterp.interp <:tactic<zify_positive>>
-       | "N" -> Tacinterp.interp <:tactic<zify_N>>
-       | "Z" -> Tacinterp.interp <:tactic<zify_op>>
-       | s -> Util.error ("No ROmega knowledge base for type "^s))
-    (Util.list_uniquize (List.sort compare l))
-  in 
-  tclTHEN
-    (tclREPEAT (tclPROGRESS (tclTHENLIST tacs)))
-    (tclTHEN 
-       (* because of the contradiction process in (r)omega, 
-          we'd better leave as little as possible in the conclusion,
-          for an easier decidability argument. *)
-       Tactics.intros 
-       total_reflexive_omega_tactic)
-
-
-TACTIC EXTEND romega
-|  [ "romega" ] -> [ romega_tactic [] ]
-END
-
-TACTIC EXTEND romega'
-| [ "romega" "with" ne_ident_list(l) ] -> 
-    [ romega_tactic (List.map Names.string_of_id l) ]
-| [ "romega" "with" "*" ] -> [ romega_tactic ["nat";"positive";"N";"Z"] ]
-END
diff --git a/contrib/romega/refl_omega.ml b/contrib/romega/refl_omega.ml
deleted file mode 100644
index fc4f7a8f..00000000
--- a/contrib/romega/refl_omega.ml
+++ /dev/null
@@ -1,1299 +0,0 @@
-(*************************************************************************
-
-   PROJET RNRT Calife - 2001
-   Author: Pierre Crégut - France Télécom R&D
-   Licence : LGPL version 2.1
-
- *************************************************************************)
-
-open Util
-open Const_omega
-module OmegaSolver = Omega.MakeOmegaSolver (Bigint)
-open OmegaSolver
-
-(* \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
-
-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 Bigint.bigint
-    (* 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: 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 *)
-(* La liste des dépendances est triée et sans redondance *)
-type solution = {
-  s_index : int;
-  s_equa_deps : int list;
-  s_trace : 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_connector_id env =  
-   env.cnt_connectors <- succ env.cnt_connectors; 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.pr_lconstr t);
-	Pp.flush_all ();
-	loop c (succ i) l in
-  print_newline ();
-  Printf.printf "  ENVIRONMENT OF PROPOSITIONS :\n\n"; loop 'P' 0 env.props;
-  Printf.printf "  ENVIRONMENT OF TERMS :\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_eq, rst_omega_eq = 
-  let cpt = ref 0 in 
-  (function () -> incr cpt; !cpt), 
-  (function () -> cpt:=0)
-
-(* generation d'identifiant de variable pour Omega *)
-
-let new_omega_var, rst_omega_var = 
-  let cpt = ref 0 in 
-  (function () -> incr cpt; !cpt), 
-  (function () -> cpt:=0)
-
-(* Affichage des variables d'un système *)
-
-let display_omega_var i = Printf.sprintf "OV%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_var () in 
-    env.om_vars <- (t,v) :: env.om_vars; v
-  end
-
-(* Ajout forcé d'un lien entre un terme et une variable  Cas où la
-   variable est créée par Omega et où il faut la lier après coup à 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_index0 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_index0 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.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 "%s" (Bigint.to_string 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 ({v=intern_omega env v; c=n} :: accu) r
-    | Oint n ->
-	let id = new_omega_eq () in
-	(*i tag_equation name id; i*)
-	{kind = kind; body = List.rev accu; 
-	 constant = n; id = id}
-    | t -> print_string "CO"; oprint stdout t; failwith "compile_equation" in
-  loop []
-
-(* \subsection{Omega vers Oformula} *)
-
-let rec oformula_of_omega env af = 
-  let rec loop = function
-    | ({v=v; c=n}::r) ->
-	Oplus(Omult(unintern_omega env v,Oint n),loop r)
-    | [] -> Oint af.constant in
-  loop af.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 Z.plus [| loop t1; loop t2 |]
-  | Oopp t -> app Z.opp [| loop t |]
-  | Omult(t1,t2) -> app Z.mult [| loop t1; loop t2 |]
-  | Oint v ->   Z.mk 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 Z.minus [| loop t1; loop t2 |] in
-  loop t
-
-(* \subsection{Oformula vers COQ reifié} *)
-
-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 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 [| Z.mk 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 [| Z.mk constant |] in
-  let mk_coeff {c=c; v=v} t =
-    let coef = 
-      app coq_t_mult 
-	[| reified_of_formula env (unintern_omega env v); 
-	   app coq_t_int [| Z.mk 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 -> 
-    display_eq display_omega_var (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. *)
-(* Chaque fonction retourne une liste triée sans redondance *)
-
-let (@@) = list_merge_uniq compare
-
-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 rec vars_of_equations = function
-  | [] -> [] 
-  | e::l -> 
-      (vars_of_formula e.e_left) @@
-      (vars_of_formula e.e_right) @@
-      (vars_of_equations l)
-
-let rec vars_of_prop = function 
-  | Pequa(_,e) -> vars_of_equations [e]
-  | Pnot p -> vars_of_prop p
-  | Por(_,p1,p2) -> (vars_of_prop p1) @@ (vars_of_prop p2)
-  | Pand(_,p1,p2) -> (vars_of_prop p1) @@ (vars_of_prop p2)
-  | Pimp(_,p1,p2) -> (vars_of_prop p1) @@ (vars_of_prop p2)
-  | Pprop _ | Ptrue | Pfalse -> []
-
-(* \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(Bigint.neg 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 (Bigint.neg 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(negone))
- | Oint i -> do_list [Lazy.force coq_c_reduce] ,Oint(Bigint.neg 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
-      (({c=c1;v=v1}::l1) as l1'),
-      (({c=c2;v=v2}::l2) as l2') ->
-	if v1 = v2 then
-          if k1*c1 + k2 * c2 = zero 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))
-    | ({c=c1;v=v1}::l1), [] -> 
-	  Lazy.force coq_f_left :: loop(l1,[])
-    | [],({c=c2;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_comm], 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_comm], 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 two)
-   | Oatom v, Omult(_,c2) -> 
-       Lazy.force coq_c_red2, Omult(Oatom v,Oplus(c2,Oint one))
-   | Omult (v1,c1),Oatom v -> 
-       Lazy.force coq_c_red3, Omult(Oatom v,Oplus(c1,Oint one))
-   | 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 one) 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{Réordonnancement} *)
-
-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) -> 
-      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 zero) in
-      tac @ [Lazy.force coq_c_red6], final
-
-(* \subsection{Elimination des zéros} *)
-
-let rec clear_zero = function
-   Oplus(Omult(Oatom v,Oint n),r) when n=zero ->
-     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_comm; 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)) EQUA
-    | Neq -> mk_step t1 t2 (fun o1 o2 -> Oplus (o1,Oopp o2)) DISE
-    | Leq -> mk_step t1 t2 (fun o1 o2 -> Oplus (o2,Oopp o1)) INEQ
-    | Geq -> mk_step t1 t2 (fun o1 o2 -> Oplus (o1,Oopp o2)) INEQ
-    | Lt  ->
-        mk_step t1 t2 (fun o1 o2 -> Oplus (Oplus(o2,Oint negone),Oopp o1))
-	  INEQ
-    | Gt ->
-        mk_step t1 t2 (fun o1 o2 -> Oplus (Oplus(o1,Oint negone),Oopp o2)) 
-	  INEQ
-  with e when Logic.catchable_exception e -> raise e
-
-(* \section{Compilation des hypothèses} *)
-
-let rec oformula_of_constr env t =
-  match Z.parse_term t with 
-    | Tplus (t1,t2) -> binop env (fun x y -> Oplus(x,y)) t1 t2
-    | Tminus (t1,t2) -> binop env (fun x y -> Ominus(x,y)) t1 t2
-    | Tmult (t1,t2) when Z.is_scalar t1 || Z.is_scalar t2 -> 
-	binop env (fun x y -> Omult(x,y)) t1 t2
-    | Topp t -> Oopp(oformula_of_constr env t)
-    | Tsucc t -> Oplus(oformula_of_constr env t, Oint one)
-    | Tnum n -> Oint n
-    | _ -> 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_connector_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 = 
-  match Z.parse_rel gl c with 
-    | Req (t1,t2) -> mk_equation env ctxt c Eq t1 t2
-    | Rne (t1,t2) -> mk_equation env ctxt c Neq t1 t2
-    | Rle (t1,t2) -> mk_equation env ctxt c Leq t1 t2
-    | Rlt (t1,t2) -> mk_equation env ctxt c Lt t1 t2
-    | Rge (t1,t2) -> mk_equation env ctxt c Geq t1 t2
-    | Rgt (t1,t2) -> mk_equation env ctxt c Gt t1 t2
-    | Rtrue -> Ptrue 
-    | Rfalse -> Pfalse
-    | Rnot t -> 
-	let t' = 
-	  oproposition_of_constr 
-	    env (not negated, depends, origin,(O_mono::path)) gl t in 
-	Pnot t'
-    | Ror (t1,t2) -> 
-	binprop env ctxt (not negated) negated gl (fun i x y -> Por(i,x,y)) t1 t2
-    | Rand (t1,t2) -> 
-	binprop env ctxt negated negated gl
-	  (fun i x y -> Pand(i,x,y)) t1 t2
-    | Rimp (t1,t2) ->
-	binprop env ctxt (not negated) (not negated) gl 
-	  (fun i x y -> Pimp(i,x,y)) t1 t2
-    | Riff (t1,t2) ->
-	binprop env ctxt negated negated gl 
-	  (fun i x y -> Pand(i,x,y)) (Term.mkArrow t1 t2) (Term.mkArrow t2 t1)
-    | _ -> 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 "REIFED PROBLEM\n\n";
-    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
-	list_cartesian (@) 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 "  E%d : %a %s 0\n"
-      om_e.id
-      (fun _ -> display_eq display_omega_var) 
-      (om_e.body, om_e.constant)
-      (operator_of_eq om_e.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\n"
-      (Names.string_of_id oformula_eq.e_origin.o_hyp)
-      (if oformula_eq.e_negated then "yes" else "no") in
-
-  let display_system syst =
-    Printf.printf "=SYSTEM===================================\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
-	| HYP e -> [e.id] @@ (hyps_used_in_trace l)
-	| 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
-	| STATE action ->
-	    (*i nlle_equa: afine, def: afine, eq_orig: afine, i*)
-            (*i coef: int, var:int                            i*)
-	    action :: variable_stated_in_trace l
-	| 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 = 
-  (* Il faut trier les variables par ordre d'introduction pour ne pas risquer
-     de définir dans le mauvais ordre *)
-  let stated_equations = 
-    let cmpvar x y = Pervasives.(-) x.st_var y.st_var in 
-    let rec loop = function
-      | Tree(_,t1,t2) -> List.merge cmpvar (loop t1) (loop t2)
-      | Leaf s -> List.sort cmpvar (variable_stated_in_trace s.s_trace) 
-    in loop tree
-  in 
-  let add_env st = 
-    (* On retransforme la définition de v en formule reifiée *)
-    let v_def = oformula_of_omega env st.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 Z.typ; 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.st_var; 
-    (v, term_to_generalize,term_to_reify,st.st_def.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 *)
-
-(* PL: experimentally, the result order of the following function seems 
-   _very_ crucial for efficiency. No idea why. Do not remove the List.rev
-   or modify the current semantics of Util.list_union (some elements of first 
-   arg, then second arg), unless you know what you're doing. *)
-
-let rec get_eclatement env = function
-    i :: r -> 
-      let l = try (get_equation env i).e_depends with Not_found -> [] in
-      list_union (List.rev l) (get_eclatement env r)
-  | [] -> []
-
-let select_smaller l = 
-  let comp (_,x) (_,y) = Pervasives.(-) (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 failwith "Exit" 
-	else x :: select l
-    | [] -> [] in
-  map_succeed (function (sol,splits) -> (sol,select splits)) systems
-
-let rec equas_of_solution_tree = function
-    Tree(_,t1,t2) -> (equas_of_solution_tree t1)@@(equas_of_solution_tree t2)
-  | Leaf s -> s.s_equa_deps
-
-(* [really_useful_prop] pushes useless props in a new Pprop variable *)
-(* Things get shorter, but may also get wrong, since a Prop is considered 
-   to be undecidable in ReflOmegaCore.concl_to_hyp, whereas for instance 
-   Pfalse is decidable. So should not be used on conclusion (??) *)
-
-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.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 (succ i) l
-	end
-    | _ :: l -> loop_id (succ i) 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_index0 (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
-      | CONTRADICTION (e1,e2) :: l -> 
-	  let trace = mk_nat (List.length e1.body) in
-	  mkApp (Lazy.force coq_s_contradiction,
-		 [| trace ; mk_nat (get_hyp env_hyp e1.id); 
-		    mk_nat (get_hyp env_hyp e2.id) |])
-      | DIVIDE_AND_APPROX (e1,e2,k,d) :: l ->
-	  mkApp (Lazy.force coq_s_div_approx,
-		 [| Z.mk k; Z.mk d; 
-		    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.id) |])
-      | NOT_EXACT_DIVIDE (e1,k) :: l ->
-	  let e2_constant = floor_div e1.constant k in
-	  let d = e1.constant - e2_constant * k in
-	  let e2_body = map_eq_linear (fun c -> c / k) e1.body in
-          mkApp (Lazy.force coq_s_not_exact_divide,
-		 [|Z.mk k; Z.mk d; 
-		   reified_of_omega env e2_body e2_constant; 
-		   mk_nat (List.length e2_body); 
-		   mk_nat (get_hyp env_hyp e1.id)|])
-      | EXACT_DIVIDE (e1,k) :: l ->
-	  let e2_body =  
-	    map_eq_linear (fun c -> c / k) e1.body in
-	  let e2_constant = floor_div e1.constant k in
-          mkApp (Lazy.force coq_s_exact_divide,
-		 [|Z.mk 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.id)|])
-      | (MERGE_EQ(e3,e1,e2)) :: l ->
-	  let n1 = get_hyp env_hyp e1.id and n2 = get_hyp env_hyp e2 in
-          mkApp (Lazy.force coq_s_merge_eq,
-		 [| mk_nat (List.length e1.body);
-		    mk_nat n1; mk_nat n2;
-		    loop (CCEqua e3:: env_hyp) l |])
-      | SUM(e3,(k1,e1),(k2,e2)) :: l ->
-	  let n1 = get_hyp env_hyp e1.id 
-	  and n2 = get_hyp env_hyp e2.id in
-	  let trace = shuffle_path k1 e1.body k2 e2.body in
-          mkApp (Lazy.force coq_s_sum,
-		 [| Z.mk k1; mk_nat n1; Z.mk k2;
-		    mk_nat n2; trace; (loop (CCEqua e3 :: env_hyp) l) |])
-      | CONSTANT_NOT_NUL(e,k) :: l -> 
-          mkApp (Lazy.force coq_s_constant_not_nul,
-		 [|  mk_nat (get_hyp env_hyp e) |])
-      | CONSTANT_NEG(e,k) :: l ->
-          mkApp (Lazy.force coq_s_constant_neg,
-		 [|  mk_nat (get_hyp env_hyp e) |])
-      | STATE {st_new_eq=new_eq; st_def =def; 
-		      st_orig=orig; st_coef=m;
-                      st_var=sigma } :: l ->
-	  let n1 = get_hyp env_hyp orig.id 
-	  and n2 = get_hyp env_hyp def.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,
-		 [| Z.mk m; trace; mk_nat n1; mk_nat n2; 
-		    loop (CCEqua new_eq.id :: env_hyp) l |])
-      |	HYP _ :: l -> loop env_hyp l
-      |	CONSTANT_NUL e :: l ->
-	  mkApp (Lazy.force coq_s_constant_nul, 
-		 [|  mk_nat (get_hyp env_hyp e) |])
-      |	NEGATE_CONTRADICT(e1,e2,true) :: l ->
-	  mkApp (Lazy.force coq_s_negate_contradict, 
-		 [|  mk_nat (get_hyp env_hyp e1.id);
-		     mk_nat (get_hyp env_hyp e2.id) |])
-      |	NEGATE_CONTRADICT(e1,e2,false) :: l ->
-	  mkApp (Lazy.force coq_s_negate_contradict_inv, 
-		 [|  mk_nat (List.length e2.body); 
-		     mk_nat (get_hyp env_hyp e1.id);
-		     mk_nat (get_hyp env_hyp e2.id) |])
-      | SPLIT_INEQ(e,(e1,l1),(e2,l2)) :: l ->
-	  let i =  get_hyp env_hyp e.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.body); mk_nat i; r1 ; r2 |])
-      |	(FORGET_C _ | FORGET _ | FORGET_I _) :: l ->
-	  loop env_hyp l
-      | (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.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 = 
-      simplify_strong 
-	(new_omega_eq,new_omega_var,display_omega_var) 
-	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;
-      display_action display_omega_var 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 = 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_uniquize (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 = 
-    let really_useful_vars = vars_of_equations equations in
-    let concl_vars = vars_of_prop (List.assoc id_concl full_reified_goal) in     
-    really_useful_vars @@ concl_vars
-  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 (et de la conclusion)
-     - 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 (succ i) l
-      |	[] -> [] in
-    loop 0 all_vars_env in
-  let env_terms_reified = mk_list (Lazy.force Z.typ) 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 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 
-       [| reified_concl;env_props_reified;env_terms_reified;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
-    let correct_index = 
-      let i = list_index0 e.e_origin.o_hyp l_hyps in 
-      (* PL: it seems that additionnally introduced hyps are in the way during 
-             normalization, hence this index shifting... *) 
-      if i=0 then 0 else Pervasives.(+) i (List.length to_introduce)
-    in 	
-    app coq_pair_step [| mk_nat correct_index; 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_vm_in_concl >>
-  (*i Alternatives to the previous line: 
-   - Normalisation without VM: 
-      Tactics.normalise_in_concl
-   - Skip the conversion check and rely directly on the QED: 
-      Tacmach.convert_concl_no_check (Lazy.force coq_True) Term.VMcast >> 
-  i*)
-  Tactics.apply (Lazy.force coq_I)
-
-let total_reflexive_omega_tactic gl = 
-  Coqlib.check_required_library ["Coq";"romega";"ROmega"];
-  rst_omega_eq (); 
-  rst_omega_var ();
-  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 display_systems systems_list;
-  resolution env full_reified_goal systems_list gl
-  with NO_CONTRADICTION -> Util.error "ROmega can't solve this system"
-
-
-(*i let tester = Tacmach.hide_atomic_tactic "TestOmega" test_tactic i*)
-
-