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-rw-r--r--theories/Arith/Arith.v10
-rw-r--r--theories/Arith/Arith_base.v10
-rw-r--r--theories/Arith/Between.v86
-rw-r--r--theories/Arith/Bool_nat.v10
-rw-r--r--theories/Arith/Compare.v10
-rw-r--r--theories/Arith/Compare_dec.v22
-rw-r--r--theories/Arith/Div2.v18
-rw-r--r--theories/Arith/EqNat.v16
-rw-r--r--theories/Arith/Euclid.v10
-rw-r--r--theories/Arith/Even.v14
-rw-r--r--theories/Arith/Factorial.v10
-rw-r--r--theories/Arith/Gt.v10
-rw-r--r--theories/Arith/Le.v28
-rw-r--r--theories/Arith/Lt.v39
-rw-r--r--theories/Arith/Max.v10
-rw-r--r--theories/Arith/Min.v10
-rw-r--r--theories/Arith/Minus.v20
-rw-r--r--theories/Arith/Mult.v34
-rw-r--r--theories/Arith/PeanoNat.v40
-rw-r--r--theories/Arith/Peano_dec.v14
-rw-r--r--theories/Arith/Plus.v24
-rw-r--r--theories/Arith/Wf_nat.v10
-rw-r--r--theories/Arith/vo.itarget22
23 files changed, 264 insertions, 213 deletions
diff --git a/theories/Arith/Arith.v b/theories/Arith/Arith.v
index 953f7e4d..1cba8faf 100644
--- a/theories/Arith/Arith.v
+++ b/theories/Arith/Arith.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Export Arith_base.
diff --git a/theories/Arith/Arith_base.v b/theories/Arith/Arith_base.v
index b378828e..e3a033a4 100644
--- a/theories/Arith/Arith_base.v
+++ b/theories/Arith/Arith_base.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Export PeanoNat.
diff --git a/theories/Arith/Between.v b/theories/Arith/Between.v
index 58d3a2b3..25d84a62 100644
--- a/theories/Arith/Between.v
+++ b/theories/Arith/Between.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Le.
@@ -16,6 +18,8 @@ Implicit Types k l p q r : nat.
Section Between.
Variables P Q : nat -> Prop.
+ (** The [between] type expresses the concept
+ [forall i: nat, k <= i < l -> P i.]. *)
Inductive between k : nat -> Prop :=
| bet_emp : between k k
| bet_S : forall l, between k l -> P l -> between k (S l).
@@ -24,7 +28,7 @@ Section Between.
Lemma bet_eq : forall k l, l = k -> between k l.
Proof.
- induction 1; auto with arith.
+ intros * ->; auto with arith.
Qed.
Hint Resolve bet_eq: arith.
@@ -37,9 +41,7 @@ Section Between.
Lemma between_Sk_l : forall k l, between k l -> S k <= l -> between (S k) l.
Proof.
- intros k l H; induction H as [|l H].
- intros; absurd (S k <= k); auto with arith.
- destruct H; auto with arith.
+ induction 1 as [|* [|]]; auto with arith.
Qed.
Hint Resolve between_Sk_l: arith.
@@ -49,6 +51,8 @@ Section Between.
induction 1; auto with arith.
Qed.
+ (** The [exists_between] type expresses the concept
+ [exists i: nat, k <= i < l /\ Q i]. *)
Inductive exists_between k : nat -> Prop :=
| exists_S : forall l, exists_between k l -> exists_between k (S l)
| exists_le : forall l, k <= l -> Q l -> exists_between k (S l).
@@ -74,32 +78,32 @@ Section Between.
Lemma in_int_intro : forall p q r, p <= r -> r < q -> in_int p q r.
Proof.
- red; auto with arith.
+ split; assumption.
Qed.
Hint Resolve in_int_intro: arith.
Lemma in_int_lt : forall p q r, in_int p q r -> p < q.
Proof.
- induction 1; intros.
- apply le_lt_trans with r; auto with arith.
+ intros * [].
+ eapply le_lt_trans; eassumption.
Qed.
Lemma in_int_p_Sq :
- forall p q r, in_int p (S q) r -> in_int p q r \/ r = q :>nat.
+ forall p q r, in_int p (S q) r -> in_int p q r \/ r = q.
Proof.
- induction 1; intros.
- elim (le_lt_or_eq r q); auto with arith.
+ intros p q r [].
+ destruct (le_lt_or_eq r q); auto with arith.
Qed.
Lemma in_int_S : forall p q r, in_int p q r -> in_int p (S q) r.
Proof.
- induction 1; auto with arith.
+ intros * []; auto with arith.
Qed.
Hint Resolve in_int_S: arith.
Lemma in_int_Sp_q : forall p q r, in_int (S p) q r -> in_int p q r.
Proof.
- induction 1; auto with arith.
+ intros * []; auto with arith.
Qed.
Hint Immediate in_int_Sp_q: arith.
@@ -107,10 +111,9 @@ Section Between.
forall k l, between k l -> forall r, in_int k l r -> P r.
Proof.
induction 1; intros.
- absurd (k < k); auto with arith.
- apply in_int_lt with r; auto with arith.
- elim (in_int_p_Sq k l r); intros; auto with arith.
- rewrite H2; trivial with arith.
+ - absurd (k < k). { auto with arith. }
+ eapply in_int_lt; eassumption.
+ - destruct (in_int_p_Sq k l r) as [| ->]; auto with arith.
Qed.
Lemma in_int_between :
@@ -120,17 +123,17 @@ Section Between.
Qed.
Lemma exists_in_int :
- forall k l, exists_between k l -> exists2 m : nat, in_int k l m & Q m.
+ forall k l, exists_between k l -> exists2 m : nat, in_int k l m & Q m.
Proof.
- induction 1.
- case IHexists_between; intros p inp Qp; exists p; auto with arith.
- exists l; auto with arith.
+ induction 1 as [* ? (p, ?, ?)|].
+ - exists p; auto with arith.
+ - exists l; auto with arith.
Qed.
Lemma in_int_exists : forall k l r, in_int k l r -> Q r -> exists_between k l.
Proof.
- destruct 1; intros.
- elim H0; auto with arith.
+ intros * (?, lt_r_l) ?.
+ induction lt_r_l; auto with arith.
Qed.
Lemma between_or_exists :
@@ -139,9 +142,11 @@ Section Between.
(forall n:nat, in_int k l n -> P n \/ Q n) ->
between k l \/ exists_between k l.
Proof.
- induction 1; intros; auto with arith.
- elim IHle; intro; auto with arith.
- elim (H0 m); auto with arith.
+ induction 1 as [|m ? IHle].
+ - auto with arith.
+ - intros P_or_Q.
+ destruct IHle; auto with arith.
+ destruct (P_or_Q m); auto with arith.
Qed.
Lemma between_not_exists :
@@ -150,14 +155,14 @@ Section Between.
(forall n:nat, in_int k l n -> P n -> ~ Q n) -> ~ exists_between k l.
Proof.
induction 1; red; intros.
- absurd (k < k); 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' l); auto with arith; intros.
- absurd (exists_between k l); auto with arith.
- apply in_int_exists with l'; auto with arith.
+ - absurd (k < k); auto with arith.
+ - absurd (Q l). { auto with arith. }
+ destruct (exists_in_int k (S l)) as (l',[],?).
+ + auto with arith.
+ + replace l with l'. { trivial. }
+ destruct (le_lt_or_eq l' l); auto with arith.
+ absurd (exists_between k l). { auto with arith. }
+ eapply in_int_exists; eauto with arith.
Qed.
Inductive P_nth (init:nat) : nat -> nat -> Prop :=
@@ -168,15 +173,16 @@ Section Between.
Lemma nth_le : forall (init:nat) l (n:nat), P_nth init l n -> init <= l.
Proof.
- induction 1; intros; auto with arith.
- apply le_trans with (S k); auto with arith.
+ induction 1.
+ - auto with arith.
+ - eapply le_trans; eauto with arith.
Qed.
Definition eventually (n:nat) := exists2 k : nat, k <= n & Q k.
Lemma event_O : eventually 0 -> Q 0.
Proof.
- induction 1; intros.
+ intros (x, ?, ?).
replace 0 with x; auto with arith.
Qed.
diff --git a/theories/Arith/Bool_nat.v b/theories/Arith/Bool_nat.v
index 602555b6..d892542e 100644
--- a/theories/Arith/Bool_nat.v
+++ b/theories/Arith/Bool_nat.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Export Compare_dec.
diff --git a/theories/Arith/Compare.v b/theories/Arith/Compare.v
index 610e9a69..6778d6a0 100644
--- a/theories/Arith/Compare.v
+++ b/theories/Arith/Compare.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Equality is decidable on [nat] *)
diff --git a/theories/Arith/Compare_dec.v b/theories/Arith/Compare_dec.v
index 976507b5..713aef85 100644
--- a/theories/Arith/Compare_dec.v
+++ b/theories/Arith/Compare_dec.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Le Lt Gt Decidable PeanoNat.
@@ -133,11 +135,11 @@ Qed.
See now [Nat.compare] and its properties.
In scope [nat_scope], the notation for [Nat.compare] is "?=" *)
-Notation nat_compare := Nat.compare (compat "8.4").
+Notation nat_compare := Nat.compare (compat "8.6").
-Notation nat_compare_spec := Nat.compare_spec (compat "8.4").
-Notation nat_compare_eq_iff := Nat.compare_eq_iff (compat "8.4").
-Notation nat_compare_S := Nat.compare_succ (compat "8.4").
+Notation nat_compare_spec := Nat.compare_spec (compat "8.6").
+Notation nat_compare_eq_iff := Nat.compare_eq_iff (compat "8.6").
+Notation nat_compare_S := Nat.compare_succ (only parsing).
Lemma nat_compare_lt n m : n<m <-> (n ?= m) = Lt.
Proof.
@@ -198,9 +200,9 @@ Qed.
See now [Nat.leb] and its properties.
In scope [nat_scope], the notation for [Nat.leb] is "<=?" *)
-Notation leb := Nat.leb (compat "8.4").
+Notation leb := Nat.leb (only parsing).
-Notation leb_iff := Nat.leb_le (compat "8.4").
+Notation leb_iff := Nat.leb_le (only parsing).
Lemma leb_iff_conv m n : (n <=? m) = false <-> m < n.
Proof.
diff --git a/theories/Arith/Div2.v b/theories/Arith/Div2.v
index 016cb85e..a5e45783 100644
--- a/theories/Arith/Div2.v
+++ b/theories/Arith/Div2.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Nota : this file is OBSOLETE, and left only for compatibility.
@@ -18,7 +20,7 @@ Implicit Type n : nat.
(** Here we define [n/2] and prove some of its properties *)
-Notation div2 := Nat.div2 (compat "8.4").
+Notation div2 := Nat.div2 (only parsing).
(** Since [div2] is recursively defined on [0], [1] and [(S (S n))], it is
useful to prove the corresponding induction principle *)
@@ -28,7 +30,7 @@ Lemma ind_0_1_SS :
P 0 -> P 1 -> (forall n, P n -> P (S (S n))) -> forall n, P n.
Proof.
intros P H0 H1 H2.
- fix 1.
+ fix ind_0_1_SS 1.
destruct n as [|[|n]].
- exact H0.
- exact H1.
@@ -84,7 +86,7 @@ Qed.
(** Properties related to the double ([2n]) *)
-Notation double := Nat.double (compat "8.4").
+Notation double := Nat.double (only parsing).
Hint Unfold double Nat.double: arith.
@@ -103,7 +105,7 @@ Hint Resolve double_S: arith.
Lemma even_odd_double n :
(even n <-> n = double (div2 n)) /\ (odd n <-> n = S (double (div2 n))).
Proof.
- revert n. fix 1. destruct n as [|[|n]].
+ revert n. fix even_odd_double 1. destruct n as [|[|n]].
- (* n = 0 *)
split; split; auto with arith. inversion 1.
- (* n = 1 *)
diff --git a/theories/Arith/EqNat.v b/theories/Arith/EqNat.v
index f998c19f..4b51dfc0 100644
--- a/theories/Arith/EqNat.v
+++ b/theories/Arith/EqNat.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import PeanoNat.
@@ -69,10 +71,10 @@ Defined.
We reuse the one already defined in module [Nat].
In scope [nat_scope], the notation "=?" can be used. *)
-Notation beq_nat := Nat.eqb (compat "8.4").
+Notation beq_nat := Nat.eqb (only parsing).
-Notation beq_nat_true_iff := Nat.eqb_eq (compat "8.4").
-Notation beq_nat_false_iff := Nat.eqb_neq (compat "8.4").
+Notation beq_nat_true_iff := Nat.eqb_eq (only parsing).
+Notation beq_nat_false_iff := Nat.eqb_neq (only parsing).
Lemma beq_nat_refl n : true = (n =? n).
Proof.
diff --git a/theories/Arith/Euclid.v b/theories/Arith/Euclid.v
index fbe98d17..29f4d3e2 100644
--- a/theories/Arith/Euclid.v
+++ b/theories/Arith/Euclid.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Mult.
diff --git a/theories/Arith/Even.v b/theories/Arith/Even.v
index 3c8c250a..a1d0e9fc 100644
--- a/theories/Arith/Even.v
+++ b/theories/Arith/Even.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Nota : this file is OBSOLETE, and left only for compatibility.
@@ -36,7 +38,7 @@ Hint Constructors odd: arith.
Lemma even_equiv : forall n, even n <-> Nat.Even n.
Proof.
- fix 1.
+ fix even_equiv 1.
destruct n as [|[|n]]; simpl.
- split; [now exists 0 | constructor].
- split.
@@ -50,7 +52,7 @@ Qed.
Lemma odd_equiv : forall n, odd n <-> Nat.Odd n.
Proof.
- fix 1.
+ fix odd_equiv 1.
destruct n as [|[|n]]; simpl.
- split.
+ inversion_clear 1.
diff --git a/theories/Arith/Factorial.v b/theories/Arith/Factorial.v
index b119bb00..22f586d7 100644
--- a/theories/Arith/Factorial.v
+++ b/theories/Arith/Factorial.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import PeanoNat Plus Mult Lt.
diff --git a/theories/Arith/Gt.v b/theories/Arith/Gt.v
index 67c94fdf..52ecf131 100644
--- a/theories/Arith/Gt.v
+++ b/theories/Arith/Gt.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Theorems about [gt] in [nat].
diff --git a/theories/Arith/Le.v b/theories/Arith/Le.v
index 0fbcec57..69626cc1 100644
--- a/theories/Arith/Le.v
+++ b/theories/Arith/Le.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Order on natural numbers.
@@ -26,17 +28,17 @@ Local Open Scope nat_scope.
(** * [le] is an order on [nat] *)
-Notation le_refl := Nat.le_refl (compat "8.4").
-Notation le_trans := Nat.le_trans (compat "8.4").
-Notation le_antisym := Nat.le_antisymm (compat "8.4").
+Notation le_refl := Nat.le_refl (only parsing).
+Notation le_trans := Nat.le_trans (only parsing).
+Notation le_antisym := Nat.le_antisymm (only parsing).
Hint Resolve le_trans: arith.
Hint Immediate le_antisym: arith.
(** * Properties of [le] w.r.t 0 *)
-Notation le_0_n := Nat.le_0_l (compat "8.4"). (* 0 <= n *)
-Notation le_Sn_0 := Nat.nle_succ_0 (compat "8.4"). (* ~ S n <= 0 *)
+Notation le_0_n := Nat.le_0_l (only parsing). (* 0 <= n *)
+Notation le_Sn_0 := Nat.nle_succ_0 (only parsing). (* ~ S n <= 0 *)
Lemma le_n_0_eq n : n <= 0 -> 0 = n.
Proof.
@@ -53,8 +55,8 @@ Proof Peano.le_n_S.
Theorem le_S_n : forall n m, S n <= S m -> n <= m.
Proof Peano.le_S_n.
-Notation le_n_Sn := Nat.le_succ_diag_r (compat "8.4"). (* n <= S n *)
-Notation le_Sn_n := Nat.nle_succ_diag_l (compat "8.4"). (* ~ S n <= n *)
+Notation le_n_Sn := Nat.le_succ_diag_r (only parsing). (* n <= S n *)
+Notation le_Sn_n := Nat.nle_succ_diag_l (only parsing). (* ~ S n <= n *)
Theorem le_Sn_le : forall n m, S n <= m -> n <= m.
Proof Nat.lt_le_incl.
@@ -65,8 +67,8 @@ Hint Immediate le_n_0_eq le_Sn_le le_S_n : arith.
(** * Properties of [le] w.r.t predecessor *)
-Notation le_pred_n := Nat.le_pred_l (compat "8.4"). (* pred n <= n *)
-Notation le_pred := Nat.pred_le_mono (compat "8.4"). (* n<=m -> pred n <= pred m *)
+Notation le_pred_n := Nat.le_pred_l (only parsing). (* pred n <= n *)
+Notation le_pred := Nat.pred_le_mono (only parsing). (* n<=m -> pred n <= pred m *)
Hint Resolve le_pred_n: arith.
diff --git a/theories/Arith/Lt.v b/theories/Arith/Lt.v
index bfc2b91a..0c7515c6 100644
--- a/theories/Arith/Lt.v
+++ b/theories/Arith/Lt.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Strict order on natural numbers.
@@ -23,7 +25,7 @@ Local Open Scope nat_scope.
(** * Irreflexivity *)
-Notation lt_irrefl := Nat.lt_irrefl (compat "8.4"). (* ~ x < x *)
+Notation lt_irrefl := Nat.lt_irrefl (only parsing). (* ~ x < x *)
Hint Resolve lt_irrefl: arith.
@@ -62,12 +64,12 @@ Hint Immediate le_not_lt lt_not_le: arith.
(** * Asymmetry *)
-Notation lt_asym := Nat.lt_asymm (compat "8.4"). (* n<m -> ~m<n *)
+Notation lt_asym := Nat.lt_asymm (only parsing). (* n<m -> ~m<n *)
(** * Order and 0 *)
-Notation lt_0_Sn := Nat.lt_0_succ (compat "8.4"). (* 0 < S n *)
-Notation lt_n_0 := Nat.nlt_0_r (compat "8.4"). (* ~ n < 0 *)
+Notation lt_0_Sn := Nat.lt_0_succ (only parsing). (* 0 < S n *)
+Notation lt_n_0 := Nat.nlt_0_r (only parsing). (* ~ n < 0 *)
Theorem neq_0_lt n : 0 <> n -> 0 < n.
Proof.
@@ -84,8 +86,8 @@ Hint Immediate neq_0_lt lt_0_neq: arith.
(** * Order and successor *)
-Notation lt_n_Sn := Nat.lt_succ_diag_r (compat "8.4"). (* n < S n *)
-Notation lt_S := Nat.lt_lt_succ_r (compat "8.4"). (* n < m -> n < S m *)
+Notation lt_n_Sn := Nat.lt_succ_diag_r (only parsing). (* n < S n *)
+Notation lt_S := Nat.lt_lt_succ_r (only parsing). (* n < m -> n < S m *)
Theorem lt_n_S n m : n < m -> S n < S m.
Proof.
@@ -107,6 +109,11 @@ Proof.
intros. symmetry. now apply Nat.lt_succ_pred with m.
Qed.
+Lemma S_pred_pos n: O < n -> n = S (pred n).
+Proof.
+ apply S_pred.
+Qed.
+
Lemma lt_pred n m : S n < m -> n < pred m.
Proof.
apply Nat.lt_succ_lt_pred.
@@ -122,28 +129,28 @@ Hint Resolve lt_pred_n_n: arith.
(** * Transitivity properties *)
-Notation lt_trans := Nat.lt_trans (compat "8.4").
-Notation lt_le_trans := Nat.lt_le_trans (compat "8.4").
-Notation le_lt_trans := Nat.le_lt_trans (compat "8.4").
+Notation lt_trans := Nat.lt_trans (only parsing).
+Notation lt_le_trans := Nat.lt_le_trans (only parsing).
+Notation le_lt_trans := Nat.le_lt_trans (only parsing).
Hint Resolve lt_trans lt_le_trans le_lt_trans: arith.
(** * Large = strict or equal *)
-Notation le_lt_or_eq_iff := Nat.lt_eq_cases (compat "8.4").
+Notation le_lt_or_eq_iff := Nat.lt_eq_cases (only parsing).
Theorem le_lt_or_eq n m : n <= m -> n < m \/ n = m.
Proof.
apply Nat.lt_eq_cases.
Qed.
-Notation lt_le_weak := Nat.lt_le_incl (compat "8.4").
+Notation lt_le_weak := Nat.lt_le_incl (only parsing).
Hint Immediate lt_le_weak: arith.
(** * Dichotomy *)
-Notation le_or_lt := Nat.le_gt_cases (compat "8.4"). (* n <= m \/ m < n *)
+Notation le_or_lt := Nat.le_gt_cases (only parsing). (* n <= m \/ m < n *)
Theorem nat_total_order n m : n <> m -> n < m \/ m < n.
Proof.
diff --git a/theories/Arith/Max.v b/theories/Arith/Max.v
index 49152549..1727fa37 100644
--- a/theories/Arith/Max.v
+++ b/theories/Arith/Max.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** THIS FILE IS DEPRECATED. Use [PeanoNat.Nat] instead. *)
diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v
index 38e59b7b..fcf0b14c 100644
--- a/theories/Arith/Min.v
+++ b/theories/Arith/Min.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** THIS FILE IS DEPRECATED. Use [PeanoNat.Nat] instead. *)
diff --git a/theories/Arith/Minus.v b/theories/Arith/Minus.v
index 1fc8f790..3bf6cd95 100644
--- a/theories/Arith/Minus.v
+++ b/theories/Arith/Minus.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Properties of subtraction between natural numbers.
@@ -46,7 +48,7 @@ Qed.
(** * Diagonal *)
-Notation minus_diag := Nat.sub_diag (compat "8.4"). (* n - n = 0 *)
+Notation minus_diag := Nat.sub_diag (only parsing). (* n - n = 0 *)
Lemma minus_diag_reverse n : 0 = n - n.
Proof.
@@ -87,13 +89,13 @@ Qed.
(** * Relation with order *)
Notation minus_le_compat_r :=
- Nat.sub_le_mono_r (compat "8.4"). (* n <= m -> n - p <= m - p. *)
+ Nat.sub_le_mono_r (only parsing). (* n <= m -> n - p <= m - p. *)
Notation minus_le_compat_l :=
- Nat.sub_le_mono_l (compat "8.4"). (* n <= m -> p - m <= p - n. *)
+ Nat.sub_le_mono_l (only parsing). (* n <= m -> p - m <= p - n. *)
-Notation le_minus := Nat.le_sub_l (compat "8.4"). (* n - m <= n *)
-Notation lt_minus := Nat.sub_lt (compat "8.4"). (* m <= n -> 0 < m -> n-m < n *)
+Notation le_minus := Nat.le_sub_l (only parsing). (* n - m <= n *)
+Notation lt_minus := Nat.sub_lt (only parsing). (* m <= n -> 0 < m -> n-m < n *)
Lemma lt_O_minus_lt n m : 0 < n - m -> m < n.
Proof.
diff --git a/theories/Arith/Mult.v b/theories/Arith/Mult.v
index a173efc1..4f4aa183 100644
--- a/theories/Arith/Mult.v
+++ b/theories/Arith/Mult.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** * Properties of multiplication.
@@ -23,35 +25,35 @@ Local Open Scope nat_scope.
(** ** Zero property *)
-Notation mult_0_l := Nat.mul_0_l (compat "8.4"). (* 0 * n = 0 *)
-Notation mult_0_r := Nat.mul_0_r (compat "8.4"). (* n * 0 = 0 *)
+Notation mult_0_l := Nat.mul_0_l (only parsing). (* 0 * n = 0 *)
+Notation mult_0_r := Nat.mul_0_r (only parsing). (* n * 0 = 0 *)
(** ** 1 is neutral *)
-Notation mult_1_l := Nat.mul_1_l (compat "8.4"). (* 1 * n = n *)
-Notation mult_1_r := Nat.mul_1_r (compat "8.4"). (* n * 1 = n *)
+Notation mult_1_l := Nat.mul_1_l (only parsing). (* 1 * n = n *)
+Notation mult_1_r := Nat.mul_1_r (only parsing). (* n * 1 = n *)
Hint Resolve mult_1_l mult_1_r: arith.
(** ** Commutativity *)
-Notation mult_comm := Nat.mul_comm (compat "8.4"). (* n * m = m * n *)
+Notation mult_comm := Nat.mul_comm (only parsing). (* n * m = m * n *)
Hint Resolve mult_comm: arith.
(** ** Distributivity *)
Notation mult_plus_distr_r :=
- Nat.mul_add_distr_r (compat "8.4"). (* (n+m)*p = n*p + m*p *)
+ Nat.mul_add_distr_r (only parsing). (* (n+m)*p = n*p + m*p *)
Notation mult_plus_distr_l :=
- Nat.mul_add_distr_l (compat "8.4"). (* n*(m+p) = n*m + n*p *)
+ Nat.mul_add_distr_l (only parsing). (* n*(m+p) = n*m + n*p *)
Notation mult_minus_distr_r :=
- Nat.mul_sub_distr_r (compat "8.4"). (* (n-m)*p = n*p - m*p *)
+ Nat.mul_sub_distr_r (only parsing). (* (n-m)*p = n*p - m*p *)
Notation mult_minus_distr_l :=
- Nat.mul_sub_distr_l (compat "8.4"). (* n*(m-p) = n*m - n*p *)
+ Nat.mul_sub_distr_l (only parsing). (* n*(m-p) = n*m - n*p *)
Hint Resolve mult_plus_distr_r: arith.
Hint Resolve mult_minus_distr_r: arith.
@@ -59,7 +61,7 @@ Hint Resolve mult_minus_distr_l: arith.
(** ** Associativity *)
-Notation mult_assoc := Nat.mul_assoc (compat "8.4"). (* n*(m*p)=n*m*p *)
+Notation mult_assoc := Nat.mul_assoc (only parsing). (* n*(m*p)=n*m*p *)
Lemma mult_assoc_reverse n m p : n * m * p = n * (m * p).
Proof.
@@ -83,8 +85,8 @@ Qed.
(** ** Multiplication and successor *)
-Notation mult_succ_l := Nat.mul_succ_l (compat "8.4"). (* S n * m = n * m + m *)
-Notation mult_succ_r := Nat.mul_succ_r (compat "8.4"). (* n * S m = n * m + n *)
+Notation mult_succ_l := Nat.mul_succ_l (only parsing). (* S n * m = n * m + m *)
+Notation mult_succ_r := Nat.mul_succ_r (only parsing). (* n * S m = n * m + n *)
(** * Compatibility with orders *)
diff --git a/theories/Arith/PeanoNat.v b/theories/Arith/PeanoNat.v
index e3240bb7..bc58995f 100644
--- a/theories/Arith/PeanoNat.v
+++ b/theories/Arith/PeanoNat.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
@@ -313,7 +315,7 @@ Import Private_Parity.
Lemma even_spec : forall n, even n = true <-> Even n.
Proof.
- fix 1.
+ fix even_spec 1.
destruct n as [|[|n]]; simpl.
- split; [ now exists 0 | trivial ].
- split; [ discriminate | intro H; elim (Even_1 H) ].
@@ -323,7 +325,7 @@ Qed.
Lemma odd_spec : forall n, odd n = true <-> Odd n.
Proof.
unfold odd.
- fix 1.
+ fix odd_spec 1.
destruct n as [|[|n]]; simpl.
- split; [ discriminate | intro H; elim (Odd_0 H) ].
- split; [ now exists 0 | trivial ].
@@ -471,7 +473,7 @@ Notation "( x | y )" := (divide x y) (at level 0) : nat_scope.
Lemma gcd_divide : forall a b, (gcd a b | a) /\ (gcd a b | b).
Proof.
- fix 1.
+ fix gcd_divide 1.
intros [|a] b; simpl.
split.
now exists 0.
@@ -500,7 +502,7 @@ Qed.
Lemma gcd_greatest : forall a b c, (c|a) -> (c|b) -> (c|gcd a b).
Proof.
- fix 1.
+ fix gcd_greatest 1.
intros [|a] b; simpl; auto.
fold (b mod (S a)).
intros c H H'. apply gcd_greatest; auto.
@@ -534,7 +536,7 @@ Qed.
Lemma le_div2 n : div2 (S n) <= n.
Proof.
revert n.
- fix 1.
+ fix le_div2 1.
destruct n; simpl; trivial. apply lt_succ_r.
destruct n; [simpl|]; trivial. now constructor.
Qed.
@@ -724,6 +726,26 @@ Definition shiftr_spec a n m (_:0<=m) := shiftr_specif a n m.
Include NExtraProp.
+(** Properties of tail-recursive addition and multiplication *)
+
+Lemma tail_add_spec n m : tail_add n m = n + m.
+Proof.
+ revert m. induction n as [|n IH]; simpl; trivial.
+ intros. now rewrite IH, add_succ_r.
+Qed.
+
+Lemma tail_addmul_spec r n m : tail_addmul r n m = r + n * m.
+Proof.
+ revert m r. induction n as [| n IH]; simpl; trivial.
+ intros. rewrite IH, tail_add_spec.
+ rewrite add_assoc. f_equal. apply add_comm.
+Qed.
+
+Lemma tail_mul_spec n m : tail_mul n m = n * m.
+Proof.
+ unfold tail_mul. now rewrite tail_addmul_spec.
+Qed.
+
End Nat.
(** Re-export notations that should be available even when
diff --git a/theories/Arith/Peano_dec.v b/theories/Arith/Peano_dec.v
index f8020a50..9a24c804 100644
--- a/theories/Arith/Peano_dec.v
+++ b/theories/Arith/Peano_dec.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Decidable PeanoNat.
@@ -19,7 +21,7 @@ Proof.
- left; exists n; auto.
Defined.
-Notation eq_nat_dec := Nat.eq_dec (compat "8.4").
+Notation eq_nat_dec := Nat.eq_dec (only parsing).
Hint Resolve O_or_S eq_nat_dec: arith.
@@ -57,4 +59,4 @@ now rewrite H.
Qed.
(** For compatibility *)
-Require Import Le Lt. \ No newline at end of file
+Require Import Le Lt.
diff --git a/theories/Arith/Plus.v b/theories/Arith/Plus.v
index 600e5e51..b8297c2d 100644
--- a/theories/Arith/Plus.v
+++ b/theories/Arith/Plus.v
@@ -1,9 +1,11 @@
- (************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
+(************************************************************************)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Properties of addition.
@@ -27,12 +29,12 @@ Local Open Scope nat_scope.
(** * Neutrality of 0, commutativity, associativity *)
-Notation plus_0_l := Nat.add_0_l (compat "8.4").
-Notation plus_0_r := Nat.add_0_r (compat "8.4").
-Notation plus_comm := Nat.add_comm (compat "8.4").
-Notation plus_assoc := Nat.add_assoc (compat "8.4").
+Notation plus_0_l := Nat.add_0_l (only parsing).
+Notation plus_0_r := Nat.add_0_r (only parsing).
+Notation plus_comm := Nat.add_comm (only parsing).
+Notation plus_assoc := Nat.add_assoc (only parsing).
-Notation plus_permute := Nat.add_shuffle3 (compat "8.4").
+Notation plus_permute := Nat.add_shuffle3 (only parsing).
Definition plus_Snm_nSm : forall n m, S n + m = n + S m :=
Peano.plus_n_Sm.
@@ -138,7 +140,7 @@ Defined.
(** * Derived properties *)
-Notation plus_permute_2_in_4 := Nat.add_shuffle1 (compat "8.4").
+Notation plus_permute_2_in_4 := Nat.add_shuffle1 (only parsing).
(** * Tail-recursive plus *)
diff --git a/theories/Arith/Wf_nat.v b/theories/Arith/Wf_nat.v
index 94bbd50a..b0228898 100644
--- a/theories/Arith/Wf_nat.v
+++ b/theories/Arith/Wf_nat.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Well-founded relations and natural numbers *)
diff --git a/theories/Arith/vo.itarget b/theories/Arith/vo.itarget
deleted file mode 100644
index 0b3d31e9..00000000
--- a/theories/Arith/vo.itarget
+++ /dev/null
@@ -1,22 +0,0 @@
-PeanoNat.vo
-Arith_base.vo
-Arith.vo
-Between.vo
-Bool_nat.vo
-Compare_dec.vo
-Compare.vo
-Div2.vo
-EqNat.vo
-Euclid.vo
-Even.vo
-Factorial.vo
-Gt.vo
-Le.vo
-Lt.vo
-Max.vo
-Minus.vo
-Min.vo
-Mult.vo
-Peano_dec.vo
-Plus.vo
-Wf_nat.vo
generated by cgit v1.2.3 (git 2.39.1) at 2024-05-19 11:19:12 +0000