Imported Upstream snapshot 8.3~beta0+13298

1 files changed, 650 insertions, 0 deletions

diff --git a/plugins/field/LegacyField_Theory.v b/plugins/field/LegacyField_Theory.v
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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(* $Id$ *)
+
+Require Import List.
+Require Import Peano_dec.
+Require Import LegacyRing.
+Require Import LegacyField_Compl.
+
+Record Field_Theory : Type :=
+  {A : Type;
+   Aplus : A -> A -> A;
+   Amult : A -> A -> A;
+   Aone : A;
+   Azero : A;
+   Aopp : A -> A;
+   Aeq : A -> A -> bool;
+   Ainv : A -> A;
+   Aminus : option (A -> A -> A);
+   Adiv : option (A -> A -> A);
+   RT : Ring_Theory Aplus Amult Aone Azero Aopp Aeq;
+   Th_inv_def : forall n:A, n <> Azero -> Amult (Ainv n) n = Aone}.
+
+(* The reflexion structure *)
+Inductive ExprA : Set :=
+  | EAzero : ExprA
+  | EAone : ExprA
+  | EAplus : ExprA -> ExprA -> ExprA
+  | EAmult : ExprA -> ExprA -> ExprA
+  | EAopp : ExprA -> ExprA
+  | EAinv : ExprA -> ExprA
+  | EAvar : nat -> ExprA.
+
+(**** Decidability of equality ****)
+
+Lemma eqExprA_O : forall e1 e2:ExprA, {e1 = e2} + {e1 <> e2}.
+Proof.
+  double induction e1 e2; try intros;
+   try (left; reflexivity) || (try (right; discriminate)).
+  elim (H1 e0); intro y; elim (H2 e); intro y0;
+   try
+    (left; rewrite y; rewrite y0; auto) ||
+      (right; red in |- *; intro; inversion H3; auto).
+  elim (H1 e0); intro y; elim (H2 e); intro y0;
+   try
+    (left; rewrite y; rewrite y0; auto) ||
+      (right; red in |- *; intro; inversion H3; auto).
+  elim (H0 e); intro y.
+  left; rewrite y; auto.
+  right; red in |- *; intro; inversion H1; auto.
+  elim (H0 e); intro y.
+  left; rewrite y; auto.
+  right; red in |- *; intro; inversion H1; auto.
+  elim (eq_nat_dec n n0); intro y.
+  left; rewrite y; auto.
+  right; red in |- *; intro; inversion H; auto.
+Defined.
+
+Definition eq_nat_dec := Eval compute in eq_nat_dec.
+Definition eqExprA := Eval compute in eqExprA_O.
+
+(**** Generation of the multiplier ****)
+
+Fixpoint mult_of_list (e:list ExprA) : ExprA :=
+  match e with
+  | nil => EAone
+  | e1 :: l1 => EAmult e1 (mult_of_list l1)
+  end.
+
+Section Theory_of_fields.
+
+Variable T : Field_Theory.
+
+Let AT := A T.
+Let AplusT := Aplus T.
+Let AmultT := Amult T.
+Let AoneT := Aone T.
+Let AzeroT := Azero T.
+Let AoppT := Aopp T.
+Let AeqT := Aeq T.
+Let AinvT := Ainv T.
+Let RTT := RT T.
+Let Th_inv_defT := Th_inv_def T.
+
+Add Legacy Abstract Ring (A T) (Aplus T) (Amult T) (Aone T) (
+ Azero T) (Aopp T) (Aeq T) (RT T).
+
+Add Legacy Abstract Ring AT AplusT AmultT AoneT AzeroT AoppT AeqT RTT.
+
+(***************************)
+(*    Lemmas to be used    *)
+(***************************)
+
+Lemma AplusT_comm : forall r1 r2:AT, AplusT r1 r2 = AplusT r2 r1.
+Proof.
+  intros; legacy ring.
+Qed.
+
+Lemma AplusT_assoc :
+ forall r1 r2 r3:AT, AplusT (AplusT r1 r2) r3 = AplusT r1 (AplusT r2 r3).
+Proof.
+  intros; legacy ring.
+Qed.
+
+Lemma AmultT_comm : forall r1 r2:AT, AmultT r1 r2 = AmultT r2 r1.
+Proof.
+  intros; legacy ring.
+Qed.
+
+Lemma AmultT_assoc :
+ forall r1 r2 r3:AT, AmultT (AmultT r1 r2) r3 = AmultT r1 (AmultT r2 r3).
+Proof.
+  intros; legacy ring.
+Qed.
+
+Lemma AplusT_Ol : forall r:AT, AplusT AzeroT r = r.
+Proof.
+  intros; legacy ring.
+Qed.
+
+Lemma AmultT_1l : forall r:AT, AmultT AoneT r = r.
+Proof.
+  intros; legacy ring.
+Qed.
+
+Lemma AplusT_AoppT_r : forall r:AT, AplusT r (AoppT r) = AzeroT.
+Proof.
+  intros; legacy ring.
+Qed.
+
+Lemma AmultT_AplusT_distr :
+ forall r1 r2 r3:AT,
+   AmultT r1 (AplusT r2 r3) = AplusT (AmultT r1 r2) (AmultT r1 r3).
+Proof.
+  intros; legacy ring.
+Qed.
+
+Lemma r_AplusT_plus : forall r r1 r2:AT, AplusT r r1 = AplusT r r2 -> r1 = r2.
+Proof.
+  intros; transitivity (AplusT (AplusT (AoppT r) r) r1).
+  legacy ring.
+  transitivity (AplusT (AplusT (AoppT r) r) r2).
+  repeat rewrite AplusT_assoc; rewrite <- H; reflexivity.
+  legacy ring.
+Qed.
+
+Lemma r_AmultT_mult :
+ forall r r1 r2:AT, AmultT r r1 = AmultT r r2 -> r <> AzeroT -> r1 = r2.
+Proof.
+  intros; transitivity (AmultT (AmultT (AinvT r) r) r1).
+  rewrite Th_inv_defT; [ symmetry  in |- *; apply AmultT_1l; auto | auto ].
+  transitivity (AmultT (AmultT (AinvT r) r) r2).
+  repeat rewrite AmultT_assoc; rewrite H; trivial.
+  rewrite Th_inv_defT; [ apply AmultT_1l; auto | auto ].
+Qed.
+
+Lemma AmultT_Or : forall r:AT, AmultT r AzeroT = AzeroT.
+Proof.
+  intro; legacy ring.
+Qed.
+
+Lemma AmultT_Ol : forall r:AT, AmultT AzeroT r = AzeroT.
+Proof.
+  intro; legacy ring.
+Qed.
+
+Lemma AmultT_1r : forall r:AT, AmultT r AoneT = r.
+Proof.
+  intro; legacy ring.
+Qed.
+
+Lemma AinvT_r : forall r:AT, r <> AzeroT -> AmultT r (AinvT r) = AoneT.
+Proof.
+  intros; rewrite AmultT_comm; apply Th_inv_defT; auto.
+Qed.
+
+Lemma Rmult_neq_0_reg :
+ forall r1 r2:AT, AmultT r1 r2 <> AzeroT -> r1 <> AzeroT /\ r2 <> AzeroT.
+Proof.
+  intros r1 r2 H; split; red in |- *; intro; apply H; rewrite H0; legacy ring.
+Qed.
+
+(************************)
+(*    Interpretation    *)
+(************************)
+
+(**** ExprA --> A ****)
+
+Fixpoint interp_ExprA (lvar:list (AT * nat)) (e:ExprA) {struct e} :
+ AT :=
+  match e with
+  | EAzero => AzeroT
+  | EAone => AoneT
+  | EAplus e1 e2 => AplusT (interp_ExprA lvar e1) (interp_ExprA lvar e2)
+  | EAmult e1 e2 => AmultT (interp_ExprA lvar e1) (interp_ExprA lvar e2)
+  | EAopp e => Aopp T (interp_ExprA lvar e)
+  | EAinv e => Ainv T (interp_ExprA lvar e)
+  | EAvar n => assoc_2nd AT nat eq_nat_dec lvar n AzeroT
+  end.
+
+(************************)
+(*    Simplification    *)
+(************************)
+
+(**** Associativity ****)
+
+Definition merge_mult :=
+  (fix merge_mult (e1:ExprA) : ExprA -> ExprA :=
+     fun e2:ExprA =>
+       match e1 with
+       | EAmult t1 t2 =>
+           match t2 with
+           | EAmult t2 t3 => EAmult t1 (EAmult t2 (merge_mult t3 e2))
+           | _ => EAmult t1 (EAmult t2 e2)
+           end
+       | _ => EAmult e1 e2
+       end).
+
+Fixpoint assoc_mult (e:ExprA) : ExprA :=
+  match e with
+  | EAmult e1 e3 =>
+      match e1 with
+      | EAmult e1 e2 =>
+          merge_mult (merge_mult (assoc_mult e1) (assoc_mult e2))
+            (assoc_mult e3)
+      | _ => EAmult e1 (assoc_mult e3)
+      end
+  | _ => e
+  end.
+
+Definition merge_plus :=
+  (fix merge_plus (e1:ExprA) : ExprA -> ExprA :=
+     fun e2:ExprA =>
+       match e1 with
+       | EAplus t1 t2 =>
+           match t2 with
+           | EAplus t2 t3 => EAplus t1 (EAplus t2 (merge_plus t3 e2))
+           | _ => EAplus t1 (EAplus t2 e2)
+           end
+       | _ => EAplus e1 e2
+       end).
+
+Fixpoint assoc (e:ExprA) : ExprA :=
+  match e with
+  | EAplus e1 e3 =>
+      match e1 with
+      | EAplus e1 e2 =>
+          merge_plus (merge_plus (assoc e1) (assoc e2)) (assoc e3)
+      | _ => EAplus (assoc_mult e1) (assoc e3)
+      end
+  | _ => assoc_mult e
+  end.
+
+Lemma merge_mult_correct1 :
+ forall (e1 e2 e3:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (merge_mult (EAmult e1 e2) e3) =
+   interp_ExprA lvar (EAmult e1 (merge_mult e2 e3)).
+Proof.
+intros e1 e2; generalize e1; generalize e2; clear e1 e2.
+simple induction e2; auto; intros.
+unfold merge_mult at 1 in |- *; fold merge_mult in |- *;
+ unfold interp_ExprA at 2 in |- *; fold interp_ExprA in |- *;
+ rewrite (H0 e e3 lvar); unfold interp_ExprA at 1 in |- *;
+ fold interp_ExprA in |- *; unfold interp_ExprA at 5 in |- *;
+ fold interp_ExprA in |- *; auto.
+Qed.
+
+Lemma merge_mult_correct :
+ forall (e1 e2:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (merge_mult e1 e2) = interp_ExprA lvar (EAmult e1 e2).
+Proof.
+simple induction e1; auto; intros.
+elim e0; try (intros; simpl in |- *; legacy ring).
+unfold interp_ExprA in H2; fold interp_ExprA in H2;
+ cut
+  (AmultT (interp_ExprA lvar e2)
+     (AmultT (interp_ExprA lvar e4)
+        (AmultT (interp_ExprA lvar e) (interp_ExprA lvar e3))) =
+   AmultT
+     (AmultT (AmultT (interp_ExprA lvar e) (interp_ExprA lvar e4))
+        (interp_ExprA lvar e2)) (interp_ExprA lvar e3)).
+intro H3; rewrite H3; rewrite <- H2; rewrite merge_mult_correct1;
+ simpl in |- *; legacy ring.
+legacy ring.
+Qed.
+
+Lemma assoc_mult_correct1 :
+ forall (e1 e2:ExprA) (lvar:list (AT * nat)),
+   AmultT (interp_ExprA lvar (assoc_mult e1))
+     (interp_ExprA lvar (assoc_mult e2)) =
+   interp_ExprA lvar (assoc_mult (EAmult e1 e2)).
+Proof.
+simple induction e1; auto; intros.
+rewrite <- (H e0 lvar); simpl in |- *; rewrite merge_mult_correct;
+ simpl in |- *; rewrite merge_mult_correct; simpl in |- *;
+ auto.
+Qed.
+
+Lemma assoc_mult_correct :
+ forall (e:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (assoc_mult e) = interp_ExprA lvar e.
+Proof.
+simple induction e; auto; intros.
+elim e0; intros.
+intros; simpl in |- *; legacy ring.
+simpl in |- *; rewrite (AmultT_1l (interp_ExprA lvar (assoc_mult e1)));
+ rewrite (AmultT_1l (interp_ExprA lvar e1)); apply H0.
+simpl in |- *; rewrite (H0 lvar); auto.
+simpl in |- *; rewrite merge_mult_correct; simpl in |- *;
+ rewrite merge_mult_correct; simpl in |- *; rewrite AmultT_assoc;
+ rewrite assoc_mult_correct1; rewrite H2; simpl in |- *;
+ rewrite <- assoc_mult_correct1 in H1; unfold interp_ExprA at 3 in H1;
+ fold interp_ExprA in H1; rewrite (H0 lvar) in H1;
+ rewrite (AmultT_comm (interp_ExprA lvar e3) (interp_ExprA lvar e1));
+ rewrite <- AmultT_assoc; rewrite H1; rewrite AmultT_assoc;
+ legacy ring.
+simpl in |- *; rewrite (H0 lvar); auto.
+simpl in |- *; rewrite (H0 lvar); auto.
+simpl in |- *; rewrite (H0 lvar); auto.
+Qed.
+
+Lemma merge_plus_correct1 :
+ forall (e1 e2 e3:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (merge_plus (EAplus e1 e2) e3) =
+   interp_ExprA lvar (EAplus e1 (merge_plus e2 e3)).
+Proof.
+intros e1 e2; generalize e1; generalize e2; clear e1 e2.
+simple induction e2; auto; intros.
+unfold merge_plus at 1 in |- *; fold merge_plus in |- *;
+ unfold interp_ExprA at 2 in |- *; fold interp_ExprA in |- *;
+ rewrite (H0 e e3 lvar); unfold interp_ExprA at 1 in |- *;
+ fold interp_ExprA in |- *; unfold interp_ExprA at 5 in |- *;
+ fold interp_ExprA in |- *; auto.
+Qed.
+
+Lemma merge_plus_correct :
+ forall (e1 e2:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (merge_plus e1 e2) = interp_ExprA lvar (EAplus e1 e2).
+Proof.
+simple induction e1; auto; intros.
+elim e0; try intros; try (simpl in |- *; legacy ring).
+unfold interp_ExprA in H2; fold interp_ExprA in H2;
+ cut
+  (AplusT (interp_ExprA lvar e2)
+     (AplusT (interp_ExprA lvar e4)
+        (AplusT (interp_ExprA lvar e) (interp_ExprA lvar e3))) =
+   AplusT
+     (AplusT (AplusT (interp_ExprA lvar e) (interp_ExprA lvar e4))
+        (interp_ExprA lvar e2)) (interp_ExprA lvar e3)).
+intro H3; rewrite H3; rewrite <- H2; rewrite merge_plus_correct1;
+ simpl in |- *; legacy ring.
+legacy ring.
+Qed.
+
+Lemma assoc_plus_correct :
+ forall (e1 e2:ExprA) (lvar:list (AT * nat)),
+   AplusT (interp_ExprA lvar (assoc e1)) (interp_ExprA lvar (assoc e2)) =
+   interp_ExprA lvar (assoc (EAplus e1 e2)).
+Proof.
+simple induction e1; auto; intros.
+rewrite <- (H e0 lvar); simpl in |- *; rewrite merge_plus_correct;
+ simpl in |- *; rewrite merge_plus_correct; simpl in |- *;
+ auto.
+Qed.
+
+Lemma assoc_correct :
+ forall (e:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (assoc e) = interp_ExprA lvar e.
+Proof.
+simple induction e; auto; intros.
+elim e0; intros.
+simpl in |- *; rewrite (H0 lvar); auto.
+simpl in |- *; rewrite (H0 lvar); auto.
+simpl in |- *; rewrite merge_plus_correct; simpl in |- *;
+ rewrite merge_plus_correct; simpl in |- *; rewrite AplusT_assoc;
+ rewrite assoc_plus_correct; rewrite H2; simpl in |- *;
+ apply
+  (r_AplusT_plus (interp_ExprA lvar (assoc e1))
+     (AplusT (interp_ExprA lvar (assoc e2))
+        (AplusT (interp_ExprA lvar e3) (interp_ExprA lvar e1)))
+     (AplusT (AplusT (interp_ExprA lvar e2) (interp_ExprA lvar e3))
+        (interp_ExprA lvar e1))); rewrite <- AplusT_assoc;
+ rewrite
+  (AplusT_comm (interp_ExprA lvar (assoc e1)) (interp_ExprA lvar (assoc e2)))
+  ; rewrite assoc_plus_correct; rewrite H1; simpl in |- *;
+ rewrite (H0 lvar);
+ rewrite <-
+  (AplusT_assoc (AplusT (interp_ExprA lvar e2) (interp_ExprA lvar e1))
+     (interp_ExprA lvar e3) (interp_ExprA lvar e1))
+  ;
+ rewrite
+  (AplusT_assoc (interp_ExprA lvar e2) (interp_ExprA lvar e1)
+     (interp_ExprA lvar e3));
+ rewrite (AplusT_comm (interp_ExprA lvar e1) (interp_ExprA lvar e3));
+ rewrite <-
+  (AplusT_assoc (interp_ExprA lvar e2) (interp_ExprA lvar e3)
+     (interp_ExprA lvar e1)); apply AplusT_comm.
+unfold assoc in |- *; fold assoc in |- *; unfold interp_ExprA in |- *;
+ fold interp_ExprA in |- *; rewrite assoc_mult_correct;
+ rewrite (H0 lvar); simpl in |- *; auto.
+simpl in |- *; rewrite (H0 lvar); auto.
+simpl in |- *; rewrite (H0 lvar); auto.
+simpl in |- *; rewrite (H0 lvar); auto.
+unfold assoc in |- *; fold assoc in |- *; unfold interp_ExprA in |- *;
+ fold interp_ExprA in |- *; rewrite assoc_mult_correct;
+ simpl in |- *; auto.
+Qed.
+
+(**** Distribution *****)
+
+Fixpoint distrib_EAopp (e:ExprA) : ExprA :=
+  match e with
+  | EAplus e1 e2 => EAplus (distrib_EAopp e1) (distrib_EAopp e2)
+  | EAmult e1 e2 => EAmult (distrib_EAopp e1) (distrib_EAopp e2)
+  | EAopp e => EAmult (EAopp EAone) (distrib_EAopp e)
+  | e => e
+  end.
+
+Definition distrib_mult_right :=
+  (fix distrib_mult_right (e1:ExprA) : ExprA -> ExprA :=
+     fun e2:ExprA =>
+       match e1 with
+       | EAplus t1 t2 =>
+           EAplus (distrib_mult_right t1 e2) (distrib_mult_right t2 e2)
+       | _ => EAmult e1 e2
+       end).
+
+Fixpoint distrib_mult_left (e1 e2:ExprA) {struct e1} : ExprA :=
+  match e1 with
+  | EAplus t1 t2 =>
+      EAplus (distrib_mult_left t1 e2) (distrib_mult_left t2 e2)
+  | _ => distrib_mult_right e2 e1
+  end.
+
+Fixpoint distrib_main (e:ExprA) : ExprA :=
+  match e with
+  | EAmult e1 e2 => distrib_mult_left (distrib_main e1) (distrib_main e2)
+  | EAplus e1 e2 => EAplus (distrib_main e1) (distrib_main e2)
+  | EAopp e => EAopp (distrib_main e)
+  | _ => e
+  end.
+
+Definition distrib (e:ExprA) : ExprA := distrib_main (distrib_EAopp e).
+
+Lemma distrib_mult_right_correct :
+ forall (e1 e2:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (distrib_mult_right e1 e2) =
+   AmultT (interp_ExprA lvar e1) (interp_ExprA lvar e2).
+Proof.
+simple induction e1; try intros; simpl in |- *; auto.
+rewrite AmultT_comm; rewrite AmultT_AplusT_distr; rewrite (H e2 lvar);
+ rewrite (H0 e2 lvar); legacy ring.
+Qed.
+
+Lemma distrib_mult_left_correct :
+ forall (e1 e2:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (distrib_mult_left e1 e2) =
+   AmultT (interp_ExprA lvar e1) (interp_ExprA lvar e2).
+Proof.
+simple induction e1; try intros; simpl in |- *.
+rewrite AmultT_Ol; rewrite distrib_mult_right_correct; simpl in |- *;
+ apply AmultT_Or.
+rewrite distrib_mult_right_correct; simpl in |- *; apply AmultT_comm.
+rewrite AmultT_comm;
+ rewrite
+  (AmultT_AplusT_distr (interp_ExprA lvar e2) (interp_ExprA lvar e)
+     (interp_ExprA lvar e0));
+ rewrite (AmultT_comm (interp_ExprA lvar e2) (interp_ExprA lvar e));
+ rewrite (AmultT_comm (interp_ExprA lvar e2) (interp_ExprA lvar e0));
+ rewrite (H e2 lvar); rewrite (H0 e2 lvar); auto.
+rewrite distrib_mult_right_correct; simpl in |- *; apply AmultT_comm.
+rewrite distrib_mult_right_correct; simpl in |- *; apply AmultT_comm.
+rewrite distrib_mult_right_correct; simpl in |- *; apply AmultT_comm.
+rewrite distrib_mult_right_correct; simpl in |- *; apply AmultT_comm.
+Qed.
+
+Lemma distrib_correct :
+ forall (e:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (distrib e) = interp_ExprA lvar e.
+Proof.
+simple induction e; intros; auto.
+simpl in |- *; rewrite <- (H lvar); rewrite <- (H0 lvar);
+ unfold distrib in |- *; simpl in |- *; auto.
+simpl in |- *; rewrite <- (H lvar); rewrite <- (H0 lvar);
+ unfold distrib in |- *; simpl in |- *; apply distrib_mult_left_correct.
+simpl in |- *; fold AoppT in |- *; rewrite <- (H lvar);
+ unfold distrib in |- *; simpl in |- *; rewrite distrib_mult_right_correct;
+ simpl in |- *; fold AoppT in |- *; legacy ring.
+Qed.
+
+(**** Multiplication by the inverse product ****)
+
+Lemma mult_eq :
+ forall (e1 e2 a:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar a <> AzeroT ->
+   interp_ExprA lvar (EAmult a e1) = interp_ExprA lvar (EAmult a e2) ->
+   interp_ExprA lvar e1 = interp_ExprA lvar e2.
+Proof.
+  simpl in |- *; intros;
+   apply
+    (r_AmultT_mult (interp_ExprA lvar a) (interp_ExprA lvar e1)
+       (interp_ExprA lvar e2)); assumption.
+Qed.
+
+Fixpoint multiply_aux (a e:ExprA) {struct e} : ExprA :=
+  match e with
+  | EAplus e1 e2 => EAplus (EAmult a e1) (multiply_aux a e2)
+  | _ => EAmult a e
+  end.
+
+Definition multiply (e:ExprA) : ExprA :=
+  match e with
+  | EAmult a e1 => multiply_aux a e1
+  | _ => e
+  end.
+
+Lemma multiply_aux_correct :
+ forall (a e:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (multiply_aux a e) =
+   AmultT (interp_ExprA lvar a) (interp_ExprA lvar e).
+Proof.
+simple induction e; simpl in |- *; intros; try rewrite merge_mult_correct;
+ auto.
+  simpl in |- *; rewrite (H0 lvar); legacy ring.
+Qed.
+
+Lemma multiply_correct :
+ forall (e:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar (multiply e) = interp_ExprA lvar e.
+Proof.
+  simple induction e; simpl in |- *; auto.
+  intros; apply multiply_aux_correct.
+Qed.
+
+(**** Permutations and simplification ****)
+
+Fixpoint monom_remove (a m:ExprA) {struct m} : ExprA :=
+  match m with
+  | EAmult m0 m1 =>
+      match eqExprA m0 (EAinv a) with
+      | left _ => m1
+      | right _ => EAmult m0 (monom_remove a m1)
+      end
+  | _ =>
+      match eqExprA m (EAinv a) with
+      | left _ => EAone
+      | right _ => EAmult a m
+      end
+  end.
+
+Definition monom_simplif_rem :=
+  (fix monom_simplif_rem (a:ExprA) : ExprA -> ExprA :=
+     fun m:ExprA =>
+       match a with
+       | EAmult a0 a1 => monom_simplif_rem a1 (monom_remove a0 m)
+       | _ => monom_remove a m
+       end).
+
+Definition monom_simplif (a m:ExprA) : ExprA :=
+  match m with
+  | EAmult a' m' =>
+      match eqExprA a a' with
+      | left _ => monom_simplif_rem a m'
+      | right _ => m
+      end
+  | _ => m
+  end.
+
+Fixpoint inverse_simplif (a e:ExprA) {struct e} : ExprA :=
+  match e with
+  | EAplus e1 e2 => EAplus (monom_simplif a e1) (inverse_simplif a e2)
+  | _ => monom_simplif a e
+  end.
+
+Lemma monom_remove_correct :
+ forall (e a:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar a <> AzeroT ->
+   interp_ExprA lvar (monom_remove a e) =
+   AmultT (interp_ExprA lvar a) (interp_ExprA lvar e).
+Proof.
+simple induction e; intros.
+simpl in |- *; case (eqExprA EAzero (EAinv a)); intros;
+ [ inversion e0 | simpl in |- *; trivial ].
+simpl in |- *; case (eqExprA EAone (EAinv a)); intros;
+ [ inversion e0 | simpl in |- *; trivial ].
+simpl in |- *; case (eqExprA (EAplus e0 e1) (EAinv a)); intros;
+ [ inversion e2 | simpl in |- *; trivial ].
+simpl in |- *; case (eqExprA e0 (EAinv a)); intros.
+rewrite e2; simpl in |- *; fold AinvT in |- *.
+rewrite <-
+ (AmultT_assoc (interp_ExprA lvar a) (AinvT (interp_ExprA lvar a))
+    (interp_ExprA lvar e1)); rewrite AinvT_r; [ legacy ring | assumption ].
+simpl in |- *; rewrite H0; auto; legacy ring.
+simpl in |- *; fold AoppT in |- *; case (eqExprA (EAopp e0) (EAinv a));
+ intros; [ inversion e1 | simpl in |- *; trivial ].
+unfold monom_remove in |- *; case (eqExprA (EAinv e0) (EAinv a)); intros.
+case (eqExprA e0 a); intros.
+rewrite e2; simpl in |- *; fold AinvT in |- *; rewrite AinvT_r; auto.
+inversion e1; simpl in |- *; exfalso; auto.
+simpl in |- *; trivial.
+unfold monom_remove in |- *; case (eqExprA (EAvar n) (EAinv a)); intros;
+ [ inversion e0 | simpl in |- *; trivial ].
+Qed.
+
+Lemma monom_simplif_rem_correct :
+ forall (a e:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar a <> AzeroT ->
+   interp_ExprA lvar (monom_simplif_rem a e) =
+   AmultT (interp_ExprA lvar a) (interp_ExprA lvar e).
+Proof.
+simple induction a; simpl in |- *; intros; try rewrite monom_remove_correct;
+ auto.
+elim (Rmult_neq_0_reg (interp_ExprA lvar e) (interp_ExprA lvar e0) H1);
+ intros.
+rewrite (H0 (monom_remove e e1) lvar H3); rewrite monom_remove_correct; auto.
+legacy ring.
+Qed.
+
+Lemma monom_simplif_correct :
+ forall (e a:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar a <> AzeroT ->
+   interp_ExprA lvar (monom_simplif a e) = interp_ExprA lvar e.
+Proof.
+simple induction e; intros; auto.
+simpl in |- *; case (eqExprA a e0); intros.
+rewrite <- e2; apply monom_simplif_rem_correct; auto.
+simpl in |- *; trivial.
+Qed.
+
+Lemma inverse_correct :
+ forall (e a:ExprA) (lvar:list (AT * nat)),
+   interp_ExprA lvar a <> AzeroT ->
+   interp_ExprA lvar (inverse_simplif a e) = interp_ExprA lvar e.
+Proof.
+simple induction e; intros; auto.
+simpl in |- *; rewrite (H0 a lvar H1); rewrite monom_simplif_correct; auto.
+unfold inverse_simplif in |- *; rewrite monom_simplif_correct; auto.
+Qed.
+
+End Theory_of_fields.
+
+(* Compatibility *)
+Notation AplusT_sym := AplusT_comm (only parsing).
+Notation AmultT_sym := AmultT_comm (only parsing).