Mise en forme des theories

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@9245 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 85 insertions, 77 deletions

diff --git a/theories/Arith/Mult.v b/theories/Arith/Mult.v
index 85977244e..ab3e65856 100644
--- a/theories/Arith/Mult.v
+++ b/theories/Arith/Mult.v
@@ -17,19 +17,38 @@ Open Local Scope nat_scope.
 
 Implicit Types m n p : nat.
 
-(** Zero property *)
+(** Theorems about multiplication in [nat]. [mult] is defined in module [Init/Peano.v]. *)
+
+(** * [nat] is a semi-ring *)
+
+(** ** Zero property *)
 
 Lemma mult_0_r : forall n, n * 0 = 0.
 Proof.
-intro; symmetry  in |- *; apply mult_n_O.
+  intro; symmetry  in |- *; apply mult_n_O.
 Qed.
 
 Lemma mult_0_l : forall n, 0 * n = 0.
 Proof.
-reflexivity.
+  reflexivity.
+Qed.
+
+(** ** 1 is neutral *)
+
+Lemma mult_1_l : forall n, 1 * n = n.
+Proof.
+  simpl in |- *; auto with arith.
 Qed.
+Hint Resolve mult_1_l: arith v62.
 
-(** Commutativity *)
+Lemma mult_1_r : forall n, n * 1 = n.
+Proof.
+  induction n; [ trivial | 
+    simpl; rewrite IHn; reflexivity].
+Qed.
+Hint Resolve mult_1_r: arith v62.
+
+(** ** Commutativity *)
 
 Lemma mult_comm : forall n m, n * m = m * n.
 Proof.
@@ -39,12 +58,12 @@ elim H; apply plus_comm.
 Qed.
 Hint Resolve mult_comm: arith v62.
 
-(** Distributivity *)
+(** ** Distributivity *)
 
 Lemma mult_plus_distr_r : forall n m p, (n + m) * p = n * p + m * p.
 Proof.
-intros; elim n; simpl in |- *; intros; auto with arith.
-elim plus_assoc; elim H; auto with arith.
+  intros; elim n; simpl in |- *; intros; auto with arith.
+  elim plus_assoc; elim H; auto with arith.
 Qed.
 Hint Resolve mult_plus_distr_r: arith v62.
 
@@ -56,9 +75,9 @@ Qed.
 
 Lemma mult_minus_distr_r : forall n m p, (n - m) * p = n * p - m * p.
 Proof.
-intros; pattern n, m in |- *; apply nat_double_ind; simpl in |- *; intros;
- auto with arith.
-elim minus_plus_simpl_l_reverse; auto with arith.
+  intros; pattern n, m in |- *; apply nat_double_ind; simpl in |- *; intros;
+    auto with arith.
+  elim minus_plus_simpl_l_reverse; auto with arith.
 Qed.
 Hint Resolve mult_minus_distr_r: arith v62.
 
@@ -69,41 +88,27 @@ Proof.
 Qed.
 Hint Resolve mult_minus_distr_l: arith v62.
 
-(** Associativity *)
+(** ** Associativity *)
 
 Lemma mult_assoc_reverse : forall n m p, n * m * p = n * (m * p).
 Proof.
-intros; elim n; intros; simpl in |- *; auto with arith.
-rewrite mult_plus_distr_r.
-elim H; auto with arith.
+  intros; elim n; intros; simpl in |- *; auto with arith.
+  rewrite mult_plus_distr_r.
+  elim H; auto with arith.
 Qed.
 Hint Resolve mult_assoc_reverse: arith v62.
 
 Lemma mult_assoc : forall n m p, n * (m * p) = n * m * p.
 Proof.
-auto with arith.
+  auto with arith.
 Qed.
 Hint Resolve mult_assoc: arith v62.
 
-(** 1 is neutral *)
-
-Lemma mult_1_l : forall n, 1 * n = n.
-Proof.
-simpl in |- *; auto with arith.
-Qed.
-Hint Resolve mult_1_l: arith v62.
-
-Lemma mult_1_r : forall n, n * 1 = n.
-Proof.
-intro; elim mult_comm; auto with arith.
-Qed.
-Hint Resolve mult_1_r: arith v62.
-
-(** Compatibility with orders *)
+(** * Compatibility with orders *)
 
 Lemma mult_O_le : forall n m, m = 0 \/ n <= m * n.
 Proof.
-induction m; simpl in |- *; auto with arith.
+  induction m; simpl in |- *; auto with arith.
 Qed.
 Hint Resolve mult_O_le: arith v62.
 
@@ -117,26 +122,27 @@ Hint Resolve mult_le_compat_l: arith.
 
 
 Lemma mult_le_compat_r : forall n m p, n <= m -> n * p <= m * p.
-intros m n p H.
-rewrite mult_comm. rewrite (mult_comm n).
-auto with arith.
+Proof.
+  intros m n p H.
+  rewrite mult_comm. rewrite (mult_comm n).
+  auto with arith.
 Qed.
 
 Lemma mult_le_compat :
- forall n m p (q:nat), n <= m -> p <= q -> n * p <= m * q.
-Proof.
-intros m n p q Hmn Hpq; induction Hmn.
-induction Hpq.
-(* m*p<=m*p *)
-apply le_n.
-(* m*p<=m*m0 -> m*p<=m*(S m0) *)
-rewrite <- mult_n_Sm; apply le_trans with (m * m0).
-assumption.
-apply le_plus_l.
-(* m*p<=m0*q -> m*p<=(S m0)*q *)
-simpl in |- *; apply le_trans with (m0 * q).
-assumption.
-apply le_plus_r.
+  forall n m p (q:nat), n <= m -> p <= q -> n * p <= m * q.
+Proof.
+  intros m n p q Hmn Hpq; induction Hmn.
+  induction Hpq.
+  (* m*p<=m*p *)
+  apply le_n.
+  (* m*p<=m*m0 -> m*p<=m*(S m0) *)
+  rewrite <- mult_n_Sm; apply le_trans with (m * m0).
+  assumption.
+  apply le_plus_l.
+  (* m*p<=m0*q -> m*p<=(S m0)*q *)
+  simpl in |- *; apply le_trans with (m0 * q).
+  assumption.
+  apply le_plus_r.
 Qed.
 
 Lemma mult_S_lt_compat_l : forall n m p, m < p -> S n * m < S n * p.
@@ -148,11 +154,12 @@ Qed.
 Hint Resolve mult_S_lt_compat_l: arith.
 
 Lemma mult_lt_compat_r : forall n m p, n < m -> 0 < p -> n * p < m * p.
-intros m n p H H0.
-induction p.
-elim (lt_irrefl _ H0).
-rewrite mult_comm.
-replace (n * S p) with (S p * n); auto with arith.
+Proof.
+  intros m n p H H0.
+  induction p.
+  elim (lt_irrefl _ H0).
+  rewrite mult_comm.
+  replace (n * S p) with (S p * n); auto with arith.
 Qed.
 
 Lemma mult_S_le_reg_l : forall n m p, S n * m <= S n * p -> m <= p.
@@ -163,27 +170,28 @@ Proof.
   apply mult_S_lt_compat_l. assumption.
 Qed.
 
-(** n|->2*n and n|->2n+1 have disjoint image *)
+(** * n|->2*n and n|->2n+1 have disjoint image *)
 
 Theorem odd_even_lem : forall p q, 2 * p + 1 <> 2 * q.
-intros p; elim p; auto.
-intros q; case q; simpl in |- *.
-red in |- *; intros; discriminate.
-intros q'; rewrite (fun x y => plus_comm x (S y)); simpl in |- *; red in |- *;
- intros; discriminate.
-intros p' H q; case q.
-simpl in |- *; red in |- *; intros; discriminate.
-intros q'; red in |- *; intros H0; case (H q').
-replace (2 * q') with (2 * S q' - 2).
-rewrite <- H0; simpl in |- *; auto.
-repeat rewrite (fun x y => plus_comm x (S y)); simpl in |- *; auto.
-simpl in |- *; repeat rewrite (fun x y => plus_comm x (S y)); simpl in |- *;
- auto.
-case q'; simpl in |- *; auto.
+Proof.
+  intros p; elim p; auto.
+  intros q; case q; simpl in |- *.
+  red in |- *; intros; discriminate.
+  intros q'; rewrite (fun x y => plus_comm x (S y)); simpl in |- *; red in |- *;
+    intros; discriminate.
+  intros p' H q; case q.
+  simpl in |- *; red in |- *; intros; discriminate.
+  intros q'; red in |- *; intros H0; case (H q').
+  replace (2 * q') with (2 * S q' - 2).
+  rewrite <- H0; simpl in |- *; auto.
+  repeat rewrite (fun x y => plus_comm x (S y)); simpl in |- *; auto.
+  simpl in |- *; repeat rewrite (fun x y => plus_comm x (S y)); simpl in |- *;
+    auto.
+  case q'; simpl in |- *; auto.
 Qed.
 
 
-(** Tail-recursive mult *)
+(** * Tail-recursive mult *)
 
 (** [tail_mult] is an alternative definition for [mult] which is 
     tail-recursive, whereas [mult] is not. This can be useful 
@@ -191,23 +199,23 @@ Qed.
 
 Fixpoint mult_acc (s:nat) m n {struct n} : nat :=
   match n with
-  | O => s
-  | S p => mult_acc (tail_plus m s) m p
+    | O => s
+    | S p => mult_acc (tail_plus m s) m p
   end.
 
 Lemma mult_acc_aux : forall n m p, m + n * p = mult_acc m p n.
 Proof.
-induction n as [| p IHp]; simpl in |- *; auto.
-intros s m; rewrite <- plus_tail_plus; rewrite <- IHp.
-rewrite <- plus_assoc_reverse; apply (f_equal2 (A1:=nat) (A2:=nat)); auto.
-rewrite plus_comm; auto.
+  induction n as [| p IHp]; simpl in |- *; auto.
+  intros s m; rewrite <- plus_tail_plus; rewrite <- IHp.
+  rewrite <- plus_assoc_reverse; apply (f_equal2 (A1:=nat) (A2:=nat)); auto.
+  rewrite plus_comm; auto.
 Qed.
 
 Definition tail_mult n m := mult_acc 0 m n.
 
 Lemma mult_tail_mult : forall n m, n * m = tail_mult n m.
 Proof.
-intros; unfold tail_mult in |- *; rewrite <- mult_acc_aux; auto.
+  intros; unfold tail_mult in |- *; rewrite <- mult_acc_aux; auto.
 Qed.
 
 (** [TailSimpl] transforms any [tail_plus] and [tail_mult] into [plus] 
@@ -215,4 +223,4 @@ Qed.
 
 Ltac tail_simpl :=
   repeat rewrite <- plus_tail_plus; repeat rewrite <- mult_tail_mult;
-   simpl in |- *. 
+    simpl in |- *.
\ No newline at end of file