diff options
Diffstat (limited to 'theories/Arith')
| -rw-r--r-- | theories/Arith/Between.v | 24 | ||||
| -rw-r--r-- | theories/Arith/EqNat.v | 4 | ||||
| -rw-r--r-- | theories/Arith/Gt.v | 16 | ||||
| -rw-r--r-- | theories/Arith/Le.v | 12 | ||||
| -rw-r--r-- | theories/Arith/Lt.v | 26 | ||||
| -rw-r--r-- | theories/Arith/Max.v | 4 | ||||
| -rw-r--r-- | theories/Arith/Minus.v | 20 | ||||
| -rw-r--r-- | theories/Arith/Mult.v | 16 | ||||
| -rw-r--r-- | theories/Arith/Peano_dec.v | 5 | ||||
| -rw-r--r-- | theories/Arith/Plus.v | 12 |
10 files changed, 69 insertions, 70 deletions
diff --git a/theories/Arith/Between.v b/theories/Arith/Between.v index f998e861..58d3a2b3 100644 --- a/theories/Arith/Between.v +++ b/theories/Arith/Between.v @@ -20,20 +20,20 @@ Section Between. | bet_emp : between k k | bet_S : forall l, between k l -> P l -> between k (S l). - Hint Constructors between: arith v62. + Hint Constructors between: arith. Lemma bet_eq : forall k l, l = k -> between k l. Proof. induction 1; auto with arith. Qed. - Hint Resolve bet_eq: arith v62. + Hint Resolve bet_eq: arith. Lemma between_le : forall k l, between k l -> k <= l. Proof. induction 1; auto with arith. Qed. - Hint Immediate between_le: arith v62. + Hint Immediate between_le: arith. Lemma between_Sk_l : forall k l, between k l -> S k <= l -> between (S k) l. Proof. @@ -41,7 +41,7 @@ Section Between. intros; absurd (S k <= k); auto with arith. destruct H; auto with arith. Qed. - Hint Resolve between_Sk_l: arith v62. + Hint Resolve between_Sk_l: arith. Lemma between_restr : forall k l (m:nat), k <= l -> l <= m -> between k m -> between l m. @@ -53,7 +53,7 @@ Section Between. | 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). - Hint Constructors exists_between: arith v62. + Hint Constructors exists_between: arith. Lemma exists_le_S : forall k l, exists_between k l -> S k <= l. Proof. @@ -62,13 +62,13 @@ Section Between. Lemma exists_lt : forall k l, exists_between k l -> k < l. Proof exists_le_S. - Hint Immediate exists_le_S exists_lt: arith v62. + Hint Immediate exists_le_S exists_lt: arith. Lemma exists_S_le : forall k l, exists_between k (S l) -> k <= l. Proof. intros; apply le_S_n; auto with arith. Qed. - Hint Immediate exists_S_le: arith v62. + Hint Immediate exists_S_le: arith. Definition in_int p q r := p <= r /\ r < q. @@ -76,7 +76,7 @@ Section Between. Proof. red; auto with arith. Qed. - Hint Resolve in_int_intro: arith v62. + Hint Resolve in_int_intro: arith. Lemma in_int_lt : forall p q r, in_int p q r -> p < q. Proof. @@ -95,13 +95,13 @@ Section Between. Proof. induction 1; auto with arith. Qed. - Hint Resolve in_int_S: arith v62. + 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. Qed. - Hint Immediate in_int_Sp_q: arith v62. + Hint Immediate in_int_Sp_q: arith. Lemma between_in_int : forall k l, between k l -> forall r, in_int k l r -> P r. @@ -183,5 +183,5 @@ Section Between. End Between. Hint Resolve nth_O bet_S bet_emp bet_eq between_Sk_l exists_S exists_le - in_int_S in_int_intro: arith v62. -Hint Immediate in_int_Sp_q exists_le_S exists_S_le: arith v62. + in_int_S in_int_intro: arith. +Hint Immediate in_int_Sp_q exists_le_S exists_S_le: arith. diff --git a/theories/Arith/EqNat.v b/theories/Arith/EqNat.v index 206fc0ab..f998c19f 100644 --- a/theories/Arith/EqNat.v +++ b/theories/Arith/EqNat.v @@ -25,7 +25,7 @@ Theorem eq_nat_refl n : eq_nat n n. Proof. induction n; simpl; auto. Qed. -Hint Resolve eq_nat_refl: arith v62. +Hint Resolve eq_nat_refl: arith. (** [eq] restricted to [nat] and [eq_nat] are equivalent *) @@ -46,7 +46,7 @@ Proof. apply eq_nat_is_eq. Qed. -Hint Immediate eq_eq_nat eq_nat_eq: arith v62. +Hint Immediate eq_eq_nat eq_nat_eq: arith. Theorem eq_nat_elim : forall n (P:nat -> Prop), P n -> forall m, eq_nat n m -> P m. diff --git a/theories/Arith/Gt.v b/theories/Arith/Gt.v index dfd57694..67c94fdf 100644 --- a/theories/Arith/Gt.v +++ b/theories/Arith/Gt.v @@ -133,14 +133,14 @@ Qed. (** * Hints *) -Hint Resolve gt_Sn_O gt_Sn_n gt_n_S : arith v62. -Hint Immediate gt_S_n gt_pred : arith v62. -Hint Resolve gt_irrefl gt_asym : arith v62. -Hint Resolve le_not_gt gt_not_le : arith v62. -Hint Immediate le_S_gt gt_S_le : arith v62. -Hint Resolve gt_le_S le_gt_S : arith v62. -Hint Resolve gt_trans_S le_gt_trans gt_le_trans: arith v62. -Hint Resolve plus_gt_compat_l: arith v62. +Hint Resolve gt_Sn_O gt_Sn_n gt_n_S : arith. +Hint Immediate gt_S_n gt_pred : arith. +Hint Resolve gt_irrefl gt_asym : arith. +Hint Resolve le_not_gt gt_not_le : arith. +Hint Immediate le_S_gt gt_S_le : arith. +Hint Resolve gt_le_S le_gt_S : arith. +Hint Resolve gt_trans_S le_gt_trans gt_le_trans: arith. +Hint Resolve plus_gt_compat_l: arith. (* begin hide *) Notation gt_O_eq := gt_0_eq (only parsing). diff --git a/theories/Arith/Le.v b/theories/Arith/Le.v index ceb91187..0fbcec57 100644 --- a/theories/Arith/Le.v +++ b/theories/Arith/Le.v @@ -30,8 +30,8 @@ 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"). -Hint Resolve le_trans: arith v62. -Hint Immediate le_antisym: arith v62. +Hint Resolve le_trans: arith. +Hint Immediate le_antisym: arith. (** * Properties of [le] w.r.t 0 *) @@ -59,16 +59,16 @@ Notation le_Sn_n := Nat.nle_succ_diag_l (compat "8.4"). (* ~ S n <= n *) Theorem le_Sn_le : forall n m, S n <= m -> n <= m. Proof Nat.lt_le_incl. -Hint Resolve le_0_n le_Sn_0: arith v62. -Hint Resolve le_n_S le_n_Sn le_Sn_n : arith v62. -Hint Immediate le_n_0_eq le_Sn_le le_S_n : arith v62. +Hint Resolve le_0_n le_Sn_0: arith. +Hint Resolve le_n_S le_n_Sn le_Sn_n : arith. +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 *) -Hint Resolve le_pred_n: arith v62. +Hint Resolve le_pred_n: arith. (** * A different elimination principle for the order on natural numbers *) diff --git a/theories/Arith/Lt.v b/theories/Arith/Lt.v index f824ee6f..bfc2b91a 100644 --- a/theories/Arith/Lt.v +++ b/theories/Arith/Lt.v @@ -25,7 +25,7 @@ Local Open Scope nat_scope. Notation lt_irrefl := Nat.lt_irrefl (compat "8.4"). (* ~ x < x *) -Hint Resolve lt_irrefl: arith v62. +Hint Resolve lt_irrefl: arith. (** * Relationship between [le] and [lt] *) @@ -44,9 +44,9 @@ Proof. apply Nat.lt_succ_r. Qed. -Hint Immediate lt_le_S: arith v62. -Hint Immediate lt_n_Sm_le: arith v62. -Hint Immediate le_lt_n_Sm: arith v62. +Hint Immediate lt_le_S: arith. +Hint Immediate lt_n_Sm_le: arith. +Hint Immediate le_lt_n_Sm: arith. Theorem le_not_lt n m : n <= m -> ~ m < n. Proof. @@ -58,7 +58,7 @@ Proof. apply Nat.lt_nge. Qed. -Hint Immediate le_not_lt lt_not_le: arith v62. +Hint Immediate le_not_lt lt_not_le: arith. (** * Asymmetry *) @@ -79,8 +79,8 @@ Proof. intros. now apply Nat.neq_sym, Nat.neq_0_lt_0. Qed. -Hint Resolve lt_0_Sn lt_n_0 : arith v62. -Hint Immediate neq_0_lt lt_0_neq: arith v62. +Hint Resolve lt_0_Sn lt_n_0 : arith. +Hint Immediate neq_0_lt lt_0_neq: arith. (** * Order and successor *) @@ -97,8 +97,8 @@ Proof. apply Nat.succ_lt_mono. Qed. -Hint Resolve lt_n_Sn lt_S lt_n_S : arith v62. -Hint Immediate lt_S_n : arith v62. +Hint Resolve lt_n_Sn lt_S lt_n_S : arith. +Hint Immediate lt_S_n : arith. (** * Predecessor *) @@ -117,8 +117,8 @@ Proof. intros. now apply Nat.lt_pred_l, Nat.neq_0_lt_0. Qed. -Hint Immediate lt_pred: arith v62. -Hint Resolve lt_pred_n_n: arith v62. +Hint Immediate lt_pred: arith. +Hint Resolve lt_pred_n_n: arith. (** * Transitivity properties *) @@ -126,7 +126,7 @@ 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"). -Hint Resolve lt_trans lt_le_trans le_lt_trans: arith v62. +Hint Resolve lt_trans lt_le_trans le_lt_trans: arith. (** * Large = strict or equal *) @@ -139,7 +139,7 @@ Qed. Notation lt_le_weak := Nat.lt_le_incl (compat "8.4"). -Hint Immediate lt_le_weak: arith v62. +Hint Immediate lt_le_weak: arith. (** * Dichotomy *) diff --git a/theories/Arith/Max.v b/theories/Arith/Max.v index 65534b2e..49152549 100644 --- a/theories/Arith/Max.v +++ b/theories/Arith/Max.v @@ -42,7 +42,7 @@ Notation max_SS := Nat.succ_max_distr (only parsing). (* end hide *) Hint Resolve - Nat.max_l Nat.max_r Nat.le_max_l Nat.le_max_r : arith v62. + Nat.max_l Nat.max_r Nat.le_max_l Nat.le_max_r : arith. Hint Resolve - Nat.min_l Nat.min_r Nat.le_min_l Nat.le_min_r : arith v62. + Nat.min_l Nat.min_r Nat.le_min_l Nat.le_min_r : arith. diff --git a/theories/Arith/Minus.v b/theories/Arith/Minus.v index bc3a318c..1fc8f790 100644 --- a/theories/Arith/Minus.v +++ b/theories/Arith/Minus.v @@ -107,13 +107,13 @@ Qed. (** * Hints *) -Hint Resolve minus_n_O: arith v62. -Hint Resolve minus_Sn_m: arith v62. -Hint Resolve minus_diag_reverse: arith v62. -Hint Resolve minus_plus_simpl_l_reverse: arith v62. -Hint Immediate plus_minus: arith v62. -Hint Resolve minus_plus: arith v62. -Hint Resolve le_plus_minus: arith v62. -Hint Resolve le_plus_minus_r: arith v62. -Hint Resolve lt_minus: arith v62. -Hint Immediate lt_O_minus_lt: arith v62. +Hint Resolve minus_n_O: arith. +Hint Resolve minus_Sn_m: arith. +Hint Resolve minus_diag_reverse: arith. +Hint Resolve minus_plus_simpl_l_reverse: arith. +Hint Immediate plus_minus: arith. +Hint Resolve minus_plus: arith. +Hint Resolve le_plus_minus: arith. +Hint Resolve le_plus_minus_r: arith. +Hint Resolve lt_minus: arith. +Hint Immediate lt_O_minus_lt: arith. diff --git a/theories/Arith/Mult.v b/theories/Arith/Mult.v index 96581243..a173efc1 100644 --- a/theories/Arith/Mult.v +++ b/theories/Arith/Mult.v @@ -31,13 +31,13 @@ Notation mult_0_r := Nat.mul_0_r (compat "8.4"). (* n * 0 = 0 *) 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 *) -Hint Resolve mult_1_l mult_1_r: arith v62. +Hint Resolve mult_1_l mult_1_r: arith. (** ** Commutativity *) Notation mult_comm := Nat.mul_comm (compat "8.4"). (* n * m = m * n *) -Hint Resolve mult_comm: arith v62. +Hint Resolve mult_comm: arith. (** ** Distributivity *) @@ -53,9 +53,9 @@ Notation mult_minus_distr_r := Notation mult_minus_distr_l := Nat.mul_sub_distr_l (compat "8.4"). (* n*(m-p) = n*m - n*p *) -Hint Resolve mult_plus_distr_r: arith v62. -Hint Resolve mult_minus_distr_r: arith v62. -Hint Resolve mult_minus_distr_l: arith v62. +Hint Resolve mult_plus_distr_r: arith. +Hint Resolve mult_minus_distr_r: arith. +Hint Resolve mult_minus_distr_l: arith. (** ** Associativity *) @@ -66,8 +66,8 @@ Proof. symmetry. apply Nat.mul_assoc. Qed. -Hint Resolve mult_assoc_reverse: arith v62. -Hint Resolve mult_assoc: arith v62. +Hint Resolve mult_assoc_reverse: arith. +Hint Resolve mult_assoc: arith. (** ** Inversion lemmas *) @@ -92,7 +92,7 @@ Lemma mult_O_le n m : m = 0 \/ n <= m * n. Proof. destruct m; [left|right]; simpl; trivial using Nat.le_add_r. Qed. -Hint Resolve mult_O_le: arith v62. +Hint Resolve mult_O_le: arith. Lemma mult_le_compat_l n m p : n <= m -> p * n <= p * m. Proof. diff --git a/theories/Arith/Peano_dec.v b/theories/Arith/Peano_dec.v index 340a7968..f8020a50 100644 --- a/theories/Arith/Peano_dec.v +++ b/theories/Arith/Peano_dec.v @@ -38,8 +38,7 @@ intros m n. generalize (eq_refl (S n)). generalize n at -1. induction (S n) as [|n0 IHn0]; try discriminate. -clear n; intros n H; injection H; clear H; intro H. -rewrite <- H; intros le_mn1 le_mn2; clear n H. +clear n; intros n [= <-] le_mn1 le_mn2. pose (def_n2 := eq_refl n0); transitivity (eq_ind _ _ le_mn2 _ def_n2). 2: reflexivity. generalize def_n2; revert le_mn1 le_mn2. @@ -50,7 +49,7 @@ destruct le_mn1; intros le_mn2; destruct le_mn2. now destruct (Nat.nle_succ_diag_l _ le_mn0). + intros def_n0; generalize le_mn1; rewrite def_n0; intros le_mn0. now destruct (Nat.nle_succ_diag_l _ le_mn0). -+ intros def_n0; injection def_n0; intros ->. ++ intros def_n0. injection def_n0 as ->. rewrite (UIP_nat _ _ def_n0 eq_refl); simpl. assert (H : le_mn1 = le_mn2). now apply IHn0. diff --git a/theories/Arith/Plus.v b/theories/Arith/Plus.v index 3b823da6..600e5e51 100644 --- a/theories/Arith/Plus.v +++ b/theories/Arith/Plus.v @@ -177,12 +177,12 @@ Proof (succ_plus_discr n 3). (** * Compatibility Hints *) -Hint Immediate plus_comm : arith v62. -Hint Resolve plus_assoc plus_assoc_reverse : arith v62. -Hint Resolve plus_le_compat_l plus_le_compat_r : arith v62. -Hint Resolve le_plus_l le_plus_r le_plus_trans : arith v62. -Hint Immediate lt_plus_trans : arith v62. -Hint Resolve plus_lt_compat_l plus_lt_compat_r : arith v62. +Hint Immediate plus_comm : arith. +Hint Resolve plus_assoc plus_assoc_reverse : arith. +Hint Resolve plus_le_compat_l plus_le_compat_r : arith. +Hint Resolve le_plus_l le_plus_r le_plus_trans : arith. +Hint Immediate lt_plus_trans : arith. +Hint Resolve plus_lt_compat_l plus_lt_compat_r : arith. (** For compatibility, we "Require" the same files as before *) |