Expérimentation de NewDestruct et parfois NewInduction

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@1880 85f007b7-540e-0410-9357-904b9bb8a0f7

14 files changed, 115 insertions, 115 deletions

diff --git a/theories/Arith/Between.v b/theories/Arith/Between.v
index ab22eca22c..e6b4446017 100755
--- a/theories/Arith/Between.v
+++ b/theories/Arith/Between.v
@@ -22,29 +22,29 @@ Hint constr_between : arith v62 := Constructors between.
 
 Lemma bet_eq : (k,l:nat)(l=k)->(between k l).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 
 Hints Resolve bet_eq : arith v62.
 
 Lemma between_le : (k,l:nat)(between k l)->(le k l).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 Hints Immediate between_le : arith v62.
 
 Lemma between_Sk_l : (k,l:nat)(between k l)->(le (S k) l)->(between (S k) l).
 Proof.
-Induction 1.
+NewInduction 1.
 Intros; Absurd (le (S k) k); Auto with arith.
-Induction 1; Auto with arith.
+Induction H; Auto with arith.
 Qed.
 Hints Resolve between_Sk_l : arith v62.
 
 Lemma between_restr : 
   (k,l,m:nat)(le k l)->(le l m)->(between k m)->(between l m).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 
 Inductive exists [k:nat] : nat -> Prop
@@ -55,7 +55,7 @@ Hint constr_exists : arith v62 := Constructors exists.
 
 Lemma exists_le_S : (k,l:nat)(exists k l)->(le (S k) l).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 
 Lemma exists_lt : (k,l:nat)(exists k l)->(lt k l).
@@ -78,55 +78,55 @@ Hints Resolve in_int_intro : arith v62.
 
 Lemma in_int_lt : (p,q,r:nat)(in_int p q r)->(lt p q).
 Proof.
-Induction 1; Intros.
+NewInduction 1; Intros.
 Apply le_lt_trans with r; Auto with arith.
 Qed.
 
 Lemma in_int_p_Sq : 
   (p,q,r:nat)(in_int p (S q) r)->((in_int p q r) \/ <nat>r=q).
 Proof.
-Induction 1; Intros.
+NewInduction 1; Intros.
 Elim (le_lt_or_eq r q); Auto with arith.
 Qed.
 
 Lemma in_int_S : (p,q,r:nat)(in_int p q r)->(in_int p (S q) r).
 Proof.
-Induction 1;Auto with arith.
+NewInduction 1;Auto with arith.
 Qed.
 Hints Resolve in_int_S : arith v62.
 
 Lemma in_int_Sp_q : (p,q,r:nat)(in_int (S p) q r)->(in_int p q r).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 Hints Immediate in_int_Sp_q : arith v62.
 
 Lemma between_in_int : (k,l:nat)(between k l)->(r:nat)(in_int k l r)->(P r).
 Proof.
-Induction 1; Intros.
+NewInduction 1; Intros.
 Absurd (lt k k); Auto with arith.
 Apply in_int_lt with r; Auto with arith.
-Elim (in_int_p_Sq k l0 r); Intros; Auto with arith.
-Rewrite H4; Trivial with arith.
+Elim (in_int_p_Sq k l r); Intros; Auto with arith.
+Rewrite H2; Trivial with arith.
 Qed.
 
 Lemma in_int_between : 
   (k,l:nat)(le k l)->((r:nat)(in_int k l r)->(P r))->(between k l).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 
 Lemma exists_in_int : 
   (k,l:nat)(exists k l)->(EX m:nat | (in_int k l m) & (Q m)).
 Proof.
-Induction 1.
-Induction 2; Intros p inp Qp; Exists p; Auto with arith.
-Intros; Exists l0; Auto with arith.
+NewInduction 1.
+Case IHexists; Intros p inp Qp; Exists p; Auto with arith.
+Exists l; Auto with arith.
 Qed.
 
 Lemma in_int_exists : (k,l,r:nat)(in_int k l r)->(Q r)->(exists k l).
 Proof.
-Induction 1; Intros.
+NewInduction 1; Intros.
 Elim H1; Auto with arith.
 Qed.
 
@@ -134,23 +134,23 @@ Lemma between_or_exists :
   (k,l:nat)(le k l)->((n:nat)(in_int k l n)->((P n)\/(Q n)))
      ->((between k l)\/(exists k l)).
 Proof.
-Induction 1; Intros; Auto with arith.
-Elim H1; Intro; Auto with arith.
-Elim (H2 m); Auto with arith.
+NewInduction 1; Intros; Auto with arith.
+Elim IHle; Intro; Auto with arith.
+Elim (H0 m); Auto with arith.
 Qed.
 
 Lemma between_not_exists : (k,l:nat)(between k l)->
      ((n:nat)(in_int k l n) -> (P n) -> ~(Q n))
      -> ~(exists k l).
 Proof.
-Induction 1; Red; Intros.
+NewInduction 1; Red; Intros.
 Absurd (lt k k); Auto with arith.
-Absurd (Q l0); Auto with arith.
-Elim (exists_in_int k (S l0)); Auto with arith; Intros l' inl' Ql'.
-Replace l0 with l'; Auto with arith.
+Absurd (Q l); Auto with arith.
+Elim (exists_in_int k (S l)); Auto with arith; Intros l' inl' Ql'.
+Replace l with l'; Auto with arith.
 Elim inl'; Intros.
-Elim (le_lt_or_eq l' l0); Auto with arith; Intros.
-Absurd (exists k l0); Auto with arith.
+Elim (le_lt_or_eq l' l); Auto with arith; Intros.
+Absurd (exists k l); Auto with arith.
 Apply in_int_exists with l'; Auto with arith.
 Qed.
 
@@ -161,7 +161,7 @@ Inductive nth [init:nat] : nat->nat->Prop
 
 Lemma nth_le : (init,l,n:nat)(nth init l n)->(le init l).
 Proof.
-Induction 1; Intros; Auto with arith.
+NewInduction 1; Intros; Auto with arith.
 Apply le_trans with (S k); Auto with arith.
 Qed.
 
@@ -169,7 +169,7 @@ Definition eventually := [n:nat](EX k:nat | (le k n) & (Q k)).
 
 Lemma event_O : (eventually O)->(Q O).
 Proof.
-Induction 1; Intros.
+NewInduction 1; Intros.
 Replace O with x; Auto with arith.
 Qed.
 
diff --git a/theories/Arith/Compare.v b/theories/Arith/Compare.v
index 172ccf568a..cc4399d2f9 100755
--- a/theories/Arith/Compare.v
+++ b/theories/Arith/Compare.v
@@ -43,7 +43,7 @@ Proof.
 Intros m n H.
 LApply (lt_le_S m n); Auto with arith.
 Intro H'; LApply (le_lt_or_eq (S m) n); Auto with arith.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Right; Exists (minus n (S (S m))); Simpl.
 Rewrite (plus_sym m (minus n (S (S m)))).
 Rewrite (plus_n_Sm (minus n (S (S m))) m).
diff --git a/theories/Arith/Compare_dec.v b/theories/Arith/Compare_dec.v
index 1397326b24..531e0a9754 100755
--- a/theories/Arith/Compare_dec.v
+++ b/theories/Arith/Compare_dec.v
@@ -14,14 +14,14 @@ Require Gt.
 Require Decidable.
 
 Theorem zerop : (n:nat){n=O}+{lt O n}.
-Destruct n; Auto with arith.
+NewDestruct n; Auto with arith.
 Save.
 
 Theorem lt_eq_lt_dec : (n,m:nat){(lt n m)}+{n=m}+{(lt m n)}.
 Proof.
-Induction n; Induction m; Auto with arith.
-Intros q H'; Elim (H q).
-Induction 1; Auto with arith.
+NewInduction n; NewInduction m; Auto with arith.
+Elim (IHn m).
+NewInduction 1; Auto with arith.
 Auto with arith.
 Qed.
 
@@ -30,11 +30,11 @@ Proof lt_eq_lt_dec.
 
 Lemma le_lt_dec : (n,m:nat) {le n m} + {lt m n}.
 Proof.
-Induction n.
+NewInduction n.
 Auto with arith.
-Induction m.
+NewInduction m.
 Auto with arith.
-Intros q H'; Elim (H q); Auto with arith.
+Elim (IHn m); Auto with arith.
 Qed.
 
 Lemma le_le_S_dec : (n,m:nat) {le n m} + {le (S m) n}.
diff --git a/theories/Arith/Div2.v b/theories/Arith/Div2.v
index 60fdc68ff6..2a087a06df 100644
--- a/theories/Arith/Div2.v
+++ b/theories/Arith/Div2.v
@@ -32,8 +32,8 @@ Intros.
 Cut (n:nat)(P n)/\(P (S n)).
 Intros. Elim (H2 n). Auto with arith.
 
-Induction n0. Auto with arith.
-Intros. Elim H2; Auto with arith.
+NewInduction n0. Auto with arith.
+Intros. Elim IHn0; Auto with arith.
 Qed.
 
 (* 0 <n  =>  n/2 < n *)
diff --git a/theories/Arith/EqNat.v b/theories/Arith/EqNat.v
index 8392f17cec..d0bd9afc6e 100755
--- a/theories/Arith/EqNat.v
+++ b/theories/Arith/EqNat.v
@@ -21,17 +21,17 @@ Fixpoint  eq_nat [n:nat] : nat -> Prop :=
           end.
 
 Theorem eq_nat_refl : (n:nat)(eq_nat n n).
-Induction n; Simpl; Auto.
+NewInduction n; Simpl; Auto.
 Qed.
 Hints Resolve eq_nat_refl : arith v62.
 
 Theorem eq_eq_nat : (n,m:nat)(n=m)->(eq_nat n m).
-Induction 1; Trivial with arith.
+NewInduction 1; Trivial with arith.
 Qed.
 Hints Immediate eq_eq_nat : arith v62.
 
 Theorem eq_nat_eq : (n,m:nat)(eq_nat n m)->(n=m).
-Induction n; Induction m; Simpl; Contradiction Orelse Auto with arith.
+NewInduction n; NewInduction m; Simpl; Contradiction Orelse Auto with arith.
 Qed.
 Hints Immediate eq_nat_eq : arith v62.
 
@@ -40,15 +40,15 @@ Intros; Replace m with n; Auto with arith.
 Qed.
 
 Theorem eq_nat_decide : (n,m:nat){(eq_nat n m)}+{~(eq_nat n m)}.
-Induction n.
-Destruct m.
+NewInduction n.
+NewDestruct m.
 Auto with arith.
-Intro; Right; Red; Trivial with arith.
-Destruct m.
+Intros; Right; Red; Trivial with arith.
+NewDestruct m.
 Right; Red; Auto with arith.
 Intros.
 Simpl.
-Apply H.
+Apply IHn.
 Defined.
 
 Fixpoint  beq_nat [n:nat] : nat -> bool :=
@@ -61,7 +61,7 @@ Fixpoint  beq_nat [n:nat] : nat -> bool :=
 
 Lemma beq_nat_refl : (x:nat)true=(beq_nat x x).
 Proof.
-  Induction x; Simpl; Auto.
+  NewInduction x; Simpl; Auto.
 Qed.
 
 Definition beq_nat_eq : (x,y:nat)true=(beq_nat x y)->x=y.
diff --git a/theories/Arith/Even.v b/theories/Arith/Even.v
index bf73ecb0a7..0f1ab85596 100644
--- a/theories/Arith/Even.v
+++ b/theories/Arith/Even.v
@@ -24,22 +24,22 @@ Hint constr_odd : arith := Constructors odd.
 
 Lemma even_or_odd : (n:nat) (even n)\/(odd n).
 Proof.
-Induction n.
+NewInduction n.
 Auto with arith.
-Intros n' H. Elim H; Auto with arith.
+Elim IHn; Auto with arith.
 Save.
 
 Lemma even_odd_dec : (n:nat) { (even n) }+{ (odd n) }.
 Proof.
-Induction n.
+NewInduction n.
 Auto with arith.
-Intros n' H. Elim H; Auto with arith.
+Elim IHn; Auto with arith.
 Save.
 
 Lemma not_even_and_odd : (n:nat) (even n) -> (odd n) -> False.
 Proof.
-Induction n.
+NewInduction n.
 Intros. Inversion H0.
-Intros. Inversion H0. Inversion H1. Auto with arith.
+Intros. Inversion H. Inversion H0. Auto with arith.
 Save.
 
diff --git a/theories/Arith/Le.v b/theories/Arith/Le.v
index be420c6721..db4494fdf5 100755
--- a/theories/Arith/Le.v
+++ b/theories/Arith/Le.v
@@ -14,12 +14,12 @@
 
 Theorem le_n_S : (n,m:nat)(le n m)->(le (S n) (S m)).
 Proof.
-  Induction 1; Auto.
+  NewInduction 1; Auto.
 Qed.
 
 Theorem le_trans : (n,m,p:nat)(le n m)->(le m p)->(le n p).
 Proof.
-  Induction 2; Auto.
+  NewInduction 2; Auto.
 Qed.
 
 Theorem le_n_Sn : (n:nat)(le n (S n)).
@@ -29,14 +29,14 @@ Qed.
 
 Theorem le_O_n : (n:nat)(le O n).
 Proof.
-  Induction n ; Auto.
+  NewInduction n ; Auto.
 Qed.
 
 Hints Resolve le_n_S le_n_Sn le_O_n le_n_S le_trans : arith v62.
 
 Theorem le_pred_n : (n:nat)(le (pred n) n).
 Proof.
-Induction n ; Auto with arith.
+NewInduction n ; Auto with arith.
 Qed.
 Hints Resolve le_pred_n : arith v62.
 
@@ -73,7 +73,7 @@ Hints Resolve le_Sn_O : arith v62.
 
 Theorem le_Sn_n : (n:nat)~(le (S n) n).
 Proof.
-Induction n; Auto with arith.
+NewInduction n; Auto with arith.
 Qed.
 Hints Resolve le_Sn_n : arith v62.
 
@@ -110,7 +110,7 @@ Lemma le_elim_rel : (P:nat->nat->Prop)
      ((p,q:nat)(le p q)->(P p q)->(P (S p) (S q)))->
      (n,m:nat)(le n m)->(P n m).
 Proof.
-Induction n; Auto with arith.
-Intros n' HRec m Le.
+NewInduction n; Auto with arith.
+Intros m Le.
 Elim Le; Auto with arith.
 Qed.
diff --git a/theories/Arith/Lt.v b/theories/Arith/Lt.v
index 7856850934..7b5e089c3a 100755
--- a/theories/Arith/Lt.v
+++ b/theories/Arith/Lt.v
@@ -50,17 +50,17 @@ Hints Resolve lt_n_n : arith v62.
 
 Lemma S_pred : (n,m:nat)(lt m n)->(n=(S (pred n))).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 
 Lemma lt_pred : (n,p:nat)(lt (S n) p)->(lt n (pred p)).
 Proof.
-Induction 1; Simpl; Auto with arith.
+NewInduction 1; Simpl; Auto with arith.
 Qed.
 Hints Immediate lt_pred : arith v62.
 
 Lemma lt_pred_n_n : (n:nat)(lt O n)->(lt (pred n) n).
-Destruct 1; Simpl; Auto with arith.
+NewDestruct 1; Simpl; Auto with arith.
 Save.
 Hints Resolve lt_pred_n_n : arith v62.
 
@@ -92,14 +92,14 @@ Hints Immediate lt_le_weak : arith v62.
 
 Theorem neq_O_lt : (n:nat)(~O=n)->(lt O n).
 Proof.
-Induction n; Auto with arith.
+NewInduction n; Auto with arith.
 Intros; Absurd O=O; Trivial with arith.
 Qed.
 Hints Immediate neq_O_lt : arith v62.
 
 Theorem lt_O_neq : (n:nat)(lt O n)->(~O=n).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 Hints Immediate lt_O_neq : arith v62.
 
@@ -107,35 +107,35 @@ Hints Immediate lt_O_neq : arith v62.
 
 Theorem lt_trans : (n,m,p:nat)(lt n m)->(lt m p)->(lt n p).
 Proof.
-Induction 2; Auto with arith.
+NewInduction 2; Auto with arith.
 Qed.
 
 Theorem lt_le_trans : (n,m,p:nat)(lt n m)->(le m p)->(lt n p).
 Proof.
-Induction 2; Auto with arith.
+NewInduction 2; Auto with arith.
 Qed.
 
 Theorem le_lt_trans : (n,m,p:nat)(le n m)->(lt m p)->(lt n p).
 Proof.
-Induction 2; Auto with arith.
+NewInduction 2; Auto with arith.
 Qed.
 
 Hints Resolve lt_trans lt_le_trans le_lt_trans : arith v62.
 
 Theorem le_lt_or_eq : (n,m:nat)(le n m)->((lt n m) \/ n=m).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 
 Theorem le_or_lt : (n,m:nat)((le n m)\/(lt m n)).
 Proof.
 Intros n m; Pattern n m; Apply nat_double_ind; Auto with arith.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 
 Theorem le_not_lt : (n,m:nat)(le n m) -> ~(lt m n).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 
 Theorem lt_not_le : (n,m:nat)(lt n m) -> ~(le m n).
@@ -146,7 +146,7 @@ Hints Immediate le_not_lt lt_not_le : arith v62.
 
 Theorem lt_not_sym : (n,m:nat)(lt n m) -> ~(lt m n).
 Proof.
-Induction 1; Auto with arith.
+NewInduction 1; Auto with arith.
 Qed.
 
 Theorem nat_total_order: (m,n: nat) ~ m = n -> (lt m n) \/ (lt n m).
diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v
index bc976fbd25..a38329c347 100755
--- a/theories/Arith/Min.v
+++ b/theories/Arith/Min.v
@@ -28,15 +28,15 @@ Qed.
 
 Lemma le_min_l : (n,m:nat)(le (min n m) n).
 Proof.
-Induction n; Intros; Simpl; Auto with arith.
+NewInduction n; Intros; Simpl; Auto with arith.
 Elim m; Intros; Simpl; Auto with arith.
 Qed.
 Hints Resolve le_min_l : arith v62.
 
 Lemma le_min_r : (n,m:nat)(le (min n m) m).
 Proof.
-Induction n; Simpl; Auto with arith.
-Induction m; Simpl; Auto with arith.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Simpl; Auto with arith.
 Qed.
 Hints Resolve le_min_r : arith v62.
 
@@ -44,7 +44,7 @@ Hints Resolve le_min_r : arith v62.
 
 Lemma min_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (min n m)).
 Proof.
-Induction n; Simpl; Auto with arith.
-Induction m; Intros; Simpl; Auto with arith.
-Pattern (min n0 n1); Apply H ; Auto with arith.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Intros; Simpl; Auto with arith.
+Pattern (min n m); Apply IHn ; Auto with arith.
 Qed.
diff --git a/theories/Arith/Minus.v b/theories/Arith/Minus.v
index 3da5916e04..f3f67dea35 100755
--- a/theories/Arith/Minus.v
+++ b/theories/Arith/Minus.v
@@ -27,19 +27,19 @@ Fixpoint minus [n:nat] : nat -> nat :=
 Lemma minus_plus_simpl : 
 	(n,m,p:nat)((minus n m)=(minus (plus p n) (plus p m))).
 Proof.
-  Induction p; Simpl; Auto with arith.
+  NewInduction p; Simpl; Auto with arith.
 Qed.
 Hints Resolve minus_plus_simpl : arith v62.
 
 Lemma minus_n_O : (n:nat)(n=(minus n O)).
 Proof.
-Induction n; Simpl; Auto with arith.
+NewInduction n; Simpl; Auto with arith.
 Qed.
 Hints Resolve minus_n_O : arith v62.
 
 Lemma minus_n_n : (n:nat)(O=(minus n n)).
 Proof.
-Induction n; Simpl; Auto with arith.
+NewInduction n; Simpl; Auto with arith.
 Qed.
 Hints Resolve minus_n_n : arith v62.
 
@@ -84,7 +84,7 @@ Intros; Absurd (lt O O); Auto with arith.
 Intros p q lepq Hp gtp.
 Elim (le_lt_or_eq O p); Auto with arith.
 Auto with arith.
-Induction 1; Elim minus_n_O; Auto with arith.
+NewInduction 1; Elim minus_n_O; Auto with arith.
 Qed.
 Hints Resolve lt_minus : arith v62.
 
@@ -96,7 +96,7 @@ Qed.
 Hints Immediate lt_O_minus_lt : arith v62.
 
 Theorem pred_of_minus : (x:nat)(pred x)=(minus x (S O)).
-Induction x; Auto with arith.
+NewInduction x; Auto with arith.
 Save.
 
 
diff --git a/theories/Arith/Mult.v b/theories/Arith/Mult.v
index 8955b1ea31..86404aca3f 100755
--- a/theories/Arith/Mult.v
+++ b/theories/Arith/Mult.v
@@ -26,8 +26,8 @@ Hints Resolve mult_plus_distr : arith v62.
 
 Lemma mult_plus_distr_r : (n,m,p:nat) (mult n (plus m p))=(plus (mult n m) (mult n p)).
 Proof.
-  Induction n. Trivial.
-  Intros. Simpl. Rewrite (H m p). Apply sym_eq. Apply plus_permute_2_in_4.
+  NewInduction n. Trivial.
+  Intros. Simpl. Rewrite (IHn m p). Apply sym_eq. Apply plus_permute_2_in_4.
 Qed.
 
 Lemma mult_minus_distr : (n,m,p:nat)((mult (minus n m) p)=(minus (mult n p) (mult m p))).
@@ -39,7 +39,7 @@ Hints Resolve mult_minus_distr : arith v62.
 
 Lemma mult_O_le : (n,m:nat)(m=O)\/(le n (mult m n)).
 Proof.
-Induction m; Simpl; Auto with arith.
+NewInduction m; Simpl; Auto with arith.
 Qed.
 Hints Resolve mult_O_le : arith v62.
 
@@ -76,16 +76,16 @@ Hints Resolve mult_n_1 : arith v62.
 
 Lemma mult_le : (m,n,p:nat) (le n p) -> (le (mult m n) (mult m p)).
 Proof.
-  Induction m. Intros. Simpl. Apply le_n.
+  NewInduction m. Intros. Simpl. Apply le_n.
   Intros. Simpl. Apply le_plus_plus. Assumption.
-  Apply H. Assumption.
+  Apply IHm. Assumption.
 Qed.
 Hints Resolve mult_le : arith.
 
 Lemma mult_lt : (m,n,p:nat) (lt n p) -> (lt (mult (S m) n) (mult (S m) p)).
 Proof.
-  Induction m. Intros. Simpl. Rewrite <- plus_n_O. Rewrite <- plus_n_O. Assumption.
-  Intros. Exact (lt_plus_plus ? ? ? ? H0 (H ? ? H0)).
+  NewInduction m. Intros. Simpl. Rewrite <- plus_n_O. Rewrite <- plus_n_O. Assumption.
+  Intros. Exact (lt_plus_plus ? ? ? ? H (IHm ? ? H)).
 Qed.
 
 Hints Resolve mult_lt : arith.
diff --git a/theories/Arith/Peano_dec.v b/theories/Arith/Peano_dec.v
index b02d5a324c..d847a060d8 100755
--- a/theories/Arith/Peano_dec.v
+++ b/theories/Arith/Peano_dec.v
@@ -12,20 +12,20 @@ Require Decidable.
 
 Theorem O_or_S : (n:nat)({m:nat|(S m)=n})+{O=n}.
 Proof.
-Induction n.
+NewInduction n.
 Auto.
-Intros p H; Left; Exists p; Auto.
+Left; Exists n; Auto.
 Qed.
 
 Theorem eq_nat_dec : (n,m:nat){n=m}+{~(n=m)}.
 Proof.
-Induction n; Induction m; Auto.
-Intros q H'; Elim (H q); Auto.
+NewInduction n; NewInduction m; Auto.
+Elim (IHn m); Auto.
 Qed.
 
 Hints Resolve O_or_S eq_nat_dec : arith.
 
 Theorem dec_eq_nat:(x,y:nat)(decidable (x=y)).
 Intros x y; Unfold decidable; Elim (eq_nat_dec x y); Auto with arith.
-Save.
+Qed.
 
diff --git a/theories/Arith/Plus.v b/theories/Arith/Plus.v
index 1b70e1512b..69bbd975a8 100755
--- a/theories/Arith/Plus.v
+++ b/theories/Arith/Plus.v
@@ -32,7 +32,7 @@ Qed.
 
 Lemma simpl_plus_l : (n,m,p:nat)((plus n m)=(plus n p))->(m=p).
 Proof.
-Induction n ; Simpl ; Auto with arith.
+NewInduction n ; Simpl ; Auto with arith.
 Qed.
 
 Lemma plus_assoc_l : (n,m,p:nat)((plus n (plus m p))=(plus (plus n m) p)).
@@ -54,18 +54,18 @@ Hints Resolve plus_assoc_r : arith v62.
 
 Lemma simpl_le_plus_l : (p,n,m:nat)(le (plus p n) (plus p m))->(le n m).
 Proof.
-Induction p; Simpl; Auto with arith.
+NewInduction p; Simpl; Auto with arith.
 Qed.
 
 Lemma le_reg_l : (n,m,p:nat)(le n m)->(le (plus p n) (plus p m)).
 Proof.
-Induction p; Simpl; Auto with arith.
+NewInduction p; Simpl; Auto with arith.
 Qed.
 Hints Resolve le_reg_l : arith v62.
 
 Lemma le_reg_r : (a,b,c:nat) (le a b)->(le (plus a c) (plus b c)).
 Proof.
-Induction 1 ; Simpl; Auto with arith.
+NewInduction 1 ; Simpl; Auto with arith.
 Qed.
 Hints Resolve le_reg_r : arith v62.
 
@@ -78,7 +78,7 @@ Qed.
 
 Lemma le_plus_l : (n,m:nat)(le n (plus n m)).
 Proof.
-Induction n; Simpl; Auto with arith.
+NewInduction n; Simpl; Auto with arith.
 Qed.
 Hints Resolve le_plus_l : arith v62.
 
@@ -96,12 +96,12 @@ Hints Resolve le_plus_trans : arith v62.
 
 Lemma simpl_lt_plus_l : (n,m,p:nat)(lt (plus p n) (plus p m))->(lt n m).
 Proof.
-Induction p; Simpl; Auto with arith.
+NewInduction p; Simpl; Auto with arith.
 Qed.
 
 Lemma lt_reg_l : (n,m,p:nat)(lt n m)->(lt (plus p n) (plus p m)).
 Proof.
-Induction p; Simpl; Auto with arith.
+NewInduction p; Simpl; Auto with arith.
 Qed.
 Hints Resolve lt_reg_l : arith v62.
 
@@ -138,15 +138,15 @@ Qed.
 
 Lemma plus_is_O : (m,n:nat) (plus m n)=O -> m=O /\ n=O.
 Proof.
-  Destruct m; Auto.
+  NewDestruct m; Auto.
   Intros. Discriminate H.
 Qed.
 
 Lemma plus_is_one : 
       (m,n:nat) (plus m n)=(S O) -> {m=O /\ n=(S O)}+{m=(S O) /\ n=O}.
 Proof.
-  Destruct m; Auto.
-  Destruct n; Auto.
+  NewDestruct m; Auto.
+  NewDestruct n; Auto.
   Intros. 
   Simpl in H. Discriminate H.
 Qed.
diff --git a/theories/Arith/Wf_nat.v b/theories/Arith/Wf_nat.v
index ff502af946..f34c97d23f 100755
--- a/theories/Arith/Wf_nat.v
+++ b/theories/Arith/Wf_nat.v
@@ -25,12 +25,12 @@ Proof.
 Red.
 Cut (n:nat)(a:A)(lt (f a) n)->(Acc A ltof a).
 Intros H a;  Apply (H (S (f a))); Auto with arith.
-Induction n.
+NewInduction n.
 Intros; Absurd (lt (f a) O); Auto with arith.
-Intros m Hm a ltSma.
+Intros a ltSma.
 Apply Acc_intro.
 Unfold ltof; Intros b ltfafb.
-Apply Hm.
+Apply IHn.
 Apply lt_le_trans with (f a); Auto with arith.
 Qed.
 
@@ -60,12 +60,12 @@ Theorem induction_ltof1
 Proof.
 Intros P F; Cut (n:nat)(a:A)(lt (f a) n)->(P a).
 Intros H a;  Apply (H (S (f a))); Auto with arith.
-Induction n.
+NewInduction n.
 Intros; Absurd (lt (f a) O); Auto with arith.
-Intros m Hm a ltSma.
+Intros a ltSma.
 Apply F.
 Unfold ltof; Intros b ltfafb.
-Apply Hm.
+Apply IHn.
 Apply lt_le_trans with (f a); Auto with arith.
 Qed. 
 
@@ -94,12 +94,12 @@ Proof.
 Red.
 Cut (n:nat)(a:A)(lt (f a) n)->(Acc A R a).
 Intros H a; Apply (H (S (f a))); Auto with arith.
-Induction n.
+NewInduction n.
 Intros; Absurd (lt (f a) O); Auto with arith.
-Intros m Hm a ltSma.
+Intros a ltSma.
 Apply Acc_intro.
 Intros b ltfafb.
-Apply Hm.
+Apply IHn.
 Apply lt_le_trans with (f a); Auto with arith.
 Save.